# HG changeset patch
# User paulson
# Date 1688399197 -3600
# Node ID 1260be3f33a4359108ef63d87d322506d2c9d782
# Parent  d4125fc10c0c2ef42494a2197a6cf689873b8b50# Parent  740b23f1138a4cb339ba1f60a04b695510ee188b
merged

diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Abstract_Metric_Spaces.thy
--- a/src/HOL/Analysis/Abstract_Metric_Spaces.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Abstract_Metric_Spaces.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -2645,7 +2645,7 @@
 
 lemma continuous_map_uniform_limit_alt:
   assumes contf: "\<forall>\<^sub>F \<xi> in F. continuous_map X mtopology (f \<xi>)"
-    and gim: "g ` (topspace X) \<subseteq> M"
+    and gim: "g \<in> topspace X \<rightarrow> M"
     and dfg: "\<And>\<epsilon>. 0 < \<epsilon> \<Longrightarrow> \<forall>\<^sub>F \<xi> in F. \<forall>x \<in> topspace X. d (f \<xi> x) (g x) < \<epsilon>"
     and nontriv: "\<not> trivial_limit F"
   shows "continuous_map X mtopology g"
@@ -2653,7 +2653,7 @@
   fix \<epsilon> :: real
   assume "\<epsilon> > 0"
   with gim dfg show "\<forall>\<^sub>F \<xi> in F. \<forall>x\<in>topspace X. g x \<in> M \<and> d (f \<xi> x) (g x) < \<epsilon>"
-    by (simp add: image_subset_iff)
+    by (simp add: Pi_iff)
 qed (use nontriv in auto)
 
 
@@ -2800,14 +2800,16 @@
 lemma completely_metrizable_space_cbox: "completely_metrizable_space (top_of_set (cbox a b))"
     using closed_closedin completely_metrizable_space_closedin completely_metrizable_space_euclidean by blast
 
+
 lemma homeomorphic_completely_metrizable_space_aux:
   assumes homXY: "X homeomorphic_space Y" and X: "completely_metrizable_space X"
   shows "completely_metrizable_space Y"
 proof -
   obtain f g where hmf: "homeomorphic_map X Y f" and hmg: "homeomorphic_map Y X g"
-    and fg: "(\<forall>x \<in> topspace X. g(f x) = x) \<and> (\<forall>y \<in> topspace Y. f(g y) = y)"
-    and fim: "f ` (topspace X) = topspace Y" and gim: "g ` (topspace Y) = topspace X"
-    by (smt (verit, best) homXY homeomorphic_imp_surjective_map homeomorphic_maps_map homeomorphic_space_def)
+    and fg: "\<And>x. x \<in> topspace X \<Longrightarrow> g(f x) = x" "\<And>y. y \<in> topspace Y \<Longrightarrow> f(g y) = y"
+    and fim: "f \<in> topspace X \<rightarrow> topspace Y" and gim: "g \<in> topspace Y \<rightarrow> topspace X"
+    using homXY
+    using homeomorphic_space_unfold by blast
   obtain M d where Md: "Metric_space M d" "Metric_space.mcomplete M d" and Xeq: "X = Metric_space.mtopology M d"
     using X by (auto simp: completely_metrizable_space_def)
   then interpret MX: Metric_space M d by metis
@@ -2816,7 +2818,7 @@
   proof
     show "(D x y = 0) \<longleftrightarrow> (x = y)" if "x \<in> topspace Y" "y \<in> topspace Y" for x y
       unfolding D_def
-      by (metis that MX.topspace_mtopology MX.zero Xeq fg gim imageI)
+      by (metis that MX.topspace_mtopology MX.zero Xeq fg gim Pi_iff)
     show "D x z \<le> D x y +D y z"
       if "x \<in> topspace Y" "y \<in> topspace Y" "z \<in> topspace Y" for x y z
       using that MX.triangle Xeq gim by (auto simp: D_def)
@@ -2828,8 +2830,8 @@
     show "Metric_space (topspace Y) D"
       using MY.Metric_space_axioms by blast
     have gball: "g ` MY.mball y r = MX.mball (g y) r" if "y \<in> topspace Y" for y r
-      using that MX.topspace_mtopology Xeq gim
-      unfolding MX.mball_def MY.mball_def by (auto simp: subset_iff image_iff D_def)
+      using that MX.topspace_mtopology Xeq gim hmg homeomorphic_imp_surjective_map
+      unfolding MX.mball_def MY.mball_def  by (fastforce simp: D_def)
     have "\<exists>r>0. MY.mball y r \<subseteq> S" if "openin Y S" and "y \<in> S" for S y
     proof -
       have "openin X (g`S)"
@@ -3783,16 +3785,16 @@
 
 definition Lipschitz_continuous_map 
   where "Lipschitz_continuous_map \<equiv> 
-      \<lambda>m1 m2 f. f ` mspace m1 \<subseteq> mspace m2 \<and>
+      \<lambda>m1 m2 f. f \<in> mspace m1 \<rightarrow> mspace m2 \<and>
         (\<exists>B. \<forall>x \<in> mspace m1. \<forall>y \<in> mspace m1. mdist m2 (f x) (f y) \<le> B * mdist m1 x y)"
 
 lemma Lipschitz_continuous_map_image: 
-  "Lipschitz_continuous_map m1 m2 f \<Longrightarrow> f ` mspace m1 \<subseteq> mspace m2"
+  "Lipschitz_continuous_map m1 m2 f \<Longrightarrow> f \<in> mspace m1 \<rightarrow> mspace m2"
   by (simp add: Lipschitz_continuous_map_def)
 
 lemma Lipschitz_continuous_map_pos:
    "Lipschitz_continuous_map m1 m2 f \<longleftrightarrow>
-    f ` mspace m1 \<subseteq> mspace m2 \<and>
+    f \<in> mspace m1 \<rightarrow> mspace m2 \<and>
         (\<exists>B>0. \<forall>x \<in> mspace m1. \<forall>y \<in> mspace m1. mdist m2 (f x) (f y) \<le> B * mdist m1 x y)"
 proof -
   have "B * mdist m1 x y \<le> (\<bar>B\<bar> + 1) * mdist m1 x y" "\<bar>B\<bar> + 1 > 0" for x y B
@@ -3805,15 +3807,14 @@
 lemma Lipschitz_continuous_map_eq:
   assumes "Lipschitz_continuous_map m1 m2 f" "\<And>x. x \<in> mspace m1 \<Longrightarrow> f x = g x"
   shows "Lipschitz_continuous_map m1 m2 g"
-  using Lipschitz_continuous_map_def assms
-  by (metis (no_types, opaque_lifting) image_subset_iff)
+  using Lipschitz_continuous_map_def assms by (simp add: Lipschitz_continuous_map_pos Pi_iff)
 
 lemma Lipschitz_continuous_map_from_submetric:
   assumes "Lipschitz_continuous_map m1 m2 f"
   shows "Lipschitz_continuous_map (submetric m1 S) m2 f"
   unfolding Lipschitz_continuous_map_def 
 proof
-  show "f ` mspace (submetric m1 S) \<subseteq> mspace m2"
+  show "f \<in> mspace (submetric m1 S) \<rightarrow> mspace m2"
     using Lipschitz_continuous_map_pos assms by fastforce
 qed (use assms in \<open>fastforce simp: Lipschitz_continuous_map_def\<close>)
 
@@ -3824,37 +3825,36 @@
 
 lemma Lipschitz_continuous_map_into_submetric:
    "Lipschitz_continuous_map m1 (submetric m2 S) f \<longleftrightarrow>
-        f ` mspace m1 \<subseteq> S \<and> Lipschitz_continuous_map m1 m2 f"
+        f \<in> mspace m1 \<rightarrow> S \<and> Lipschitz_continuous_map m1 m2 f"
   by (auto simp: Lipschitz_continuous_map_def)
 
 lemma Lipschitz_continuous_map_const:
   "Lipschitz_continuous_map m1 m2 (\<lambda>x. c) \<longleftrightarrow>
         mspace m1 = {} \<or> c \<in> mspace m2"
-  unfolding Lipschitz_continuous_map_def image_subset_iff
+  unfolding Lipschitz_continuous_map_def Pi_iff
   by (metis all_not_in_conv mdist_nonneg mdist_zero mult_1)
 
 lemma Lipschitz_continuous_map_id:
    "Lipschitz_continuous_map m1 m1 (\<lambda>x. x)"
-  by (metis Lipschitz_continuous_map_def image_ident mult_1 order_refl)
+  unfolding Lipschitz_continuous_map_def by (metis funcset_id mult_1 order_refl)
 
 lemma Lipschitz_continuous_map_compose:
   assumes f: "Lipschitz_continuous_map m1 m2 f" and g: "Lipschitz_continuous_map m2 m3 g"
   shows "Lipschitz_continuous_map m1 m3 (g \<circ> f)"
-  unfolding Lipschitz_continuous_map_def image_subset_iff
+  unfolding Lipschitz_continuous_map_def 
 proof
-  show "\<forall>x\<in>mspace m1. (g \<circ> f) x \<in> mspace m3"
-    by (metis Lipschitz_continuous_map_def assms comp_apply image_subset_iff)
+  show "g \<circ> f \<in> mspace m1 \<rightarrow> mspace m3"
+    by (smt (verit, best) Lipschitz_continuous_map_image Pi_iff comp_apply f g)
   obtain B where B: "\<forall>x\<in>mspace m1. \<forall>y\<in>mspace m1. mdist m2 (f x) (f y) \<le> B * mdist m1 x y"
     using assms unfolding Lipschitz_continuous_map_def by presburger
   obtain C where "C>0" and C: "\<forall>x\<in>mspace m2. \<forall>y\<in>mspace m2. mdist m3 (g x) (g y) \<le> C * mdist m2 x y"
     using assms unfolding Lipschitz_continuous_map_pos by metis
   show "\<exists>B. \<forall>x\<in>mspace m1. \<forall>y\<in>mspace m1. mdist m3 ((g \<circ> f) x) ((g \<circ> f) y) \<le> B * mdist m1 x y"
-    apply (rule_tac x="C*B" in exI)
-  proof clarify
+  proof (intro strip exI)
     fix x y
     assume \<section>: "x \<in> mspace m1" "y \<in> mspace m1"
     then have "mdist m3 ((g \<circ> f) x) ((g \<circ> f) y) \<le> C * mdist m2 (f x) (f y)"
-      by (metis C Lipschitz_continuous_map_def f comp_apply image_subset_iff)
+      using C Lipschitz_continuous_map_image f by fastforce
     also have "\<dots> \<le> C * B * mdist m1 x y"
       by (simp add: "\<section>" B \<open>0 < C\<close>)
     finally show "mdist m3 ((g \<circ> f) x) ((g \<circ> f) y) \<le> C * B * mdist m1 x y" .
@@ -3865,12 +3865,12 @@
 
 definition uniformly_continuous_map 
   where "uniformly_continuous_map \<equiv> 
-      \<lambda>m1 m2 f. f ` mspace m1 \<subseteq> mspace m2 \<and>
+      \<lambda>m1 m2 f. f \<in> mspace m1 \<rightarrow> mspace m2 \<and>
         (\<forall>\<epsilon>>0. \<exists>\<delta>>0. \<forall>x \<in> mspace m1. \<forall>y \<in> mspace m1. 
                            mdist m1 y x < \<delta> \<longrightarrow> mdist m2 (f y) (f x) < \<epsilon>)"
 
-lemma uniformly_continuous_map_image: 
-  "uniformly_continuous_map m1 m2 f \<Longrightarrow> f ` mspace m1 \<subseteq> mspace m2"
+lemma uniformly_continuous_map_funspace: 
+  "uniformly_continuous_map m1 m2 f \<Longrightarrow> f \<in> mspace m1 \<rightarrow> mspace m2"
   by (simp add: uniformly_continuous_map_def)
 
 lemma ucmap_A:
@@ -3898,7 +3898,7 @@
   assumes \<section>: "\<And>\<rho> \<sigma> \<epsilon>. \<lbrakk>\<epsilon> > 0; range \<rho> \<subseteq> mspace m1; range \<sigma> \<subseteq> mspace m1;
                        ((\<lambda>n. mdist m1 (\<rho> n) (\<sigma> n)) \<longlonglongrightarrow> 0)\<rbrakk>
                       \<Longrightarrow> \<exists>n. mdist m2 (f (\<rho> n)) (f (\<sigma> n)) < \<epsilon>"
-    and fim: "f ` mspace m1 \<subseteq> mspace m2"
+    and fim: "f \<in> mspace m1 \<rightarrow> mspace m2"
   shows "uniformly_continuous_map m1 m2 f"
 proof -
   {assume "\<not> (\<forall>\<epsilon>>0. \<exists>\<delta>>0. \<forall>x\<in>mspace m1. \<forall>y\<in>mspace m1. mdist m1 y x < \<delta> \<longrightarrow> mdist m2 (f y) (f x) < \<epsilon>)" 
@@ -3921,13 +3921,13 @@
 
 lemma uniformly_continuous_map_sequentially:
   "uniformly_continuous_map m1 m2 f \<longleftrightarrow>
-    f ` mspace m1 \<subseteq> mspace m2 \<and>
+    f \<in> mspace m1 \<rightarrow> mspace m2 \<and>
     (\<forall>\<rho> \<sigma>. range \<rho> \<subseteq> mspace m1 \<and> range \<sigma> \<subseteq> mspace m1 \<and> (\<lambda>n. mdist m1 (\<rho> n) (\<sigma> n)) \<longlonglongrightarrow> 0
           \<longrightarrow> (\<lambda>n. mdist m2 (f (\<rho> n)) (f (\<sigma> n))) \<longlonglongrightarrow> 0)"
    (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   show "?lhs \<Longrightarrow> ?rhs"
-    by (simp add: ucmap_A uniformly_continuous_map_image)
+    by (simp add: ucmap_A uniformly_continuous_map_funspace)
   show "?rhs \<Longrightarrow> ?lhs"
     by (intro ucmap_B ucmap_C) auto
 qed
@@ -3935,14 +3935,14 @@
 
 lemma uniformly_continuous_map_sequentially_alt:
     "uniformly_continuous_map m1 m2 f \<longleftrightarrow>
-      f ` mspace m1 \<subseteq> mspace m2 \<and>
+      f \<in> mspace m1 \<rightarrow> mspace m2 \<and>
       (\<forall>\<epsilon>>0. \<forall>\<rho> \<sigma>. range \<rho> \<subseteq> mspace m1 \<and> range \<sigma> \<subseteq> mspace m1 \<and>
             ((\<lambda>n. mdist m1 (\<rho> n) (\<sigma> n)) \<longlonglongrightarrow> 0)
             \<longrightarrow> (\<exists>n. mdist m2 (f (\<rho> n)) (f (\<sigma> n)) < \<epsilon>))"
    (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   show "?lhs \<Longrightarrow> ?rhs"
-    using uniformly_continuous_map_image by (intro conjI strip ucmap_B | force simp: ucmap_A)+
+    using uniformly_continuous_map_funspace by (intro conjI strip ucmap_B | fastforce simp: ucmap_A)+
   show "?rhs \<Longrightarrow> ?lhs"
     by (intro ucmap_C) auto
 qed
@@ -3951,14 +3951,14 @@
 lemma uniformly_continuous_map_eq:
    "\<lbrakk>\<And>x. x \<in> mspace m1 \<Longrightarrow> f x = g x; uniformly_continuous_map m1 m2 f\<rbrakk>
       \<Longrightarrow> uniformly_continuous_map m1 m2 g"
-  by (simp add: uniformly_continuous_map_def)
+  by (simp add: uniformly_continuous_map_def Pi_iff)
 
 lemma uniformly_continuous_map_from_submetric:
   assumes "uniformly_continuous_map m1 m2 f"
   shows "uniformly_continuous_map (submetric m1 S) m2 f"
   unfolding uniformly_continuous_map_def 
 proof
-  show "f ` mspace (submetric m1 S) \<subseteq> mspace m2"
+  show "f \<in> mspace (submetric m1 S) \<rightarrow> mspace m2"
     using assms by (auto simp: uniformly_continuous_map_def)
 qed (use assms in \<open>force simp: uniformly_continuous_map_def\<close>)
 
@@ -3969,23 +3969,23 @@
 
 lemma uniformly_continuous_map_into_submetric:
    "uniformly_continuous_map m1 (submetric m2 S) f \<longleftrightarrow>
-        f ` mspace m1 \<subseteq> S \<and> uniformly_continuous_map m1 m2 f"
+        f \<in> mspace m1 \<rightarrow> S \<and> uniformly_continuous_map m1 m2 f"
   by (auto simp: uniformly_continuous_map_def)
 
 lemma uniformly_continuous_map_const:
   "uniformly_continuous_map m1 m2 (\<lambda>x. c) \<longleftrightarrow>
         mspace m1 = {} \<or> c \<in> mspace m2"
-  unfolding uniformly_continuous_map_def image_subset_iff
+  unfolding uniformly_continuous_map_def Pi_iff
   by (metis empty_iff equals0I mdist_zero)
 
 lemma uniformly_continuous_map_id [simp]:
    "uniformly_continuous_map m1 m1 (\<lambda>x. x)"
-  by (metis image_ident order_refl uniformly_continuous_map_def)
+  by (metis funcset_id uniformly_continuous_map_def)
 
 lemma uniformly_continuous_map_compose:
   assumes f: "uniformly_continuous_map m1 m2 f" and g: "uniformly_continuous_map m2 m3 g"
   shows "uniformly_continuous_map m1 m3 (g \<circ> f)"
-  by (smt (verit, ccfv_threshold) comp_apply f g image_subset_iff uniformly_continuous_map_def)
+  by (smt (verit, ccfv_threshold) comp_apply f g Pi_iff uniformly_continuous_map_def)
 
 lemma uniformly_continuous_map_real_const [simp]:
    "uniformly_continuous_map m euclidean_metric (\<lambda>x. c)"
@@ -4009,9 +4009,9 @@
      = Cauchy_continuous_on S f"
   by (auto simp add: Cauchy_continuous_map_def Cauchy_continuous_on_def Cauchy_def Met_TC.subspace Metric_space.MCauchy_def)
 
-lemma Cauchy_continuous_map_image:
+lemma Cauchy_continuous_map_funspace:
   assumes "Cauchy_continuous_map m1 m2 f"
-  shows "f ` mspace m1 \<subseteq> mspace m2"
+  shows "f \<in> mspace m1 \<rightarrow> mspace m2"
 proof clarsimp
   fix x
   assume "x \<in> mspace m1"
@@ -4042,14 +4042,14 @@
 
 lemma Cauchy_continuous_map_into_submetric:
    "Cauchy_continuous_map m1 (submetric m2 S) f \<longleftrightarrow>
-   f ` mspace m1 \<subseteq> S \<and> Cauchy_continuous_map m1 m2 f"  (is "?lhs \<longleftrightarrow> ?rhs")
+   f \<in> mspace m1 \<rightarrow> S \<and> Cauchy_continuous_map m1 m2 f"  (is "?lhs \<longleftrightarrow> ?rhs")
 proof -
   have "?lhs \<Longrightarrow> Cauchy_continuous_map m1 m2 f"
     by (simp add: Cauchy_continuous_map_def Metric_space.MCauchy_def [OF Metric_space_mspace_mdist])
   moreover have "?rhs \<Longrightarrow> ?lhs"
-    by (bestsimp simp add: Cauchy_continuous_map_def  Metric_space.MCauchy_def [OF Metric_space_mspace_mdist])
+    by (auto simp: Cauchy_continuous_map_def  Metric_space.MCauchy_def [OF Metric_space_mspace_mdist])
   ultimately show ?thesis
-    by (metis Cauchy_continuous_map_image le_inf_iff mspace_submetric)
+    by (metis Cauchy_continuous_map_funspace Int_iff funcsetI funcset_mem mspace_submetric)
 qed
 
 lemma Cauchy_continuous_map_const [simp]:
@@ -4060,7 +4060,7 @@
   moreover have "c \<in> mspace m2 \<Longrightarrow> Cauchy_continuous_map m1 m2 (\<lambda>x. c)"
     by (simp add: Cauchy_continuous_map_def o_def Metric_space.MCauchy_const [OF Metric_space_mspace_mdist])
   ultimately show ?thesis
-    using Cauchy_continuous_map_image by blast
+    using Cauchy_continuous_map_funspace by blast
 qed
 
 lemma Cauchy_continuous_map_id [simp]:
@@ -4076,7 +4076,7 @@
   assumes "Lipschitz_continuous_map m1 m2 f"
   shows "uniformly_continuous_map m1 m2 f"
   proof -
-  have "f ` mspace m1 \<subseteq> mspace m2"
+  have "f \<in> mspace m1 \<rightarrow> mspace m2"
     by (simp add: Lipschitz_continuous_map_image assms)
   moreover have "\<exists>\<delta>>0. \<forall>x\<in>mspace m1. \<forall>y\<in>mspace m1. mdist m1 y x < \<delta> \<longrightarrow> mdist m2 (f y) (f x) < \<epsilon>"
     if "\<epsilon> > 0" for \<epsilon>
@@ -4097,7 +4097,7 @@
    "uniformly_continuous_map m1 m2 f \<Longrightarrow> Cauchy_continuous_map m1 m2 f"
   unfolding uniformly_continuous_map_def Cauchy_continuous_map_def
   apply (simp add: image_subset_iff o_def Metric_space.MCauchy_def [OF Metric_space_mspace_mdist])
-  by meson
+  by (metis funcset_mem)
 
 lemma locally_Cauchy_continuous_map:
   assumes "\<epsilon> > 0"
@@ -4137,7 +4137,7 @@
     then have "\<sigma> n \<in> mball_of m1 (\<sigma> n) \<epsilon>"
       by (simp add: Metric_space.centre_in_mball_iff Metric_space_mspace_mdist assms(1) mball_of_def)
     then show "(f \<circ> \<sigma>) n \<in> mspace m2"
-      using Cauchy_continuous_map_image [OF \<section> [of "\<sigma> n"]] \<open>\<sigma> n \<in> mspace m1\<close> by auto
+      using Cauchy_continuous_map_funspace [OF \<section> [of "\<sigma> n"]] \<open>\<sigma> n \<in> mspace m1\<close> by auto
   qed
 qed
 
@@ -4154,7 +4154,7 @@
     unfolding limit_atin_sequentially
   proof (intro conjI strip)
     show "f x \<in> M2"
-      using Cauchy_continuous_map_image \<open>x \<in> M1\<close> assms by fastforce
+      using Cauchy_continuous_map_funspace \<open>x \<in> M1\<close> assms by fastforce
     fix \<sigma>
     assume "range \<sigma> \<subseteq> M1 - {x} \<and> limitin M1.mtopology \<sigma> x sequentially"
     then have "M1.MCauchy (\<lambda>n. if even n then \<sigma> (n div 2) else x)"
@@ -4221,8 +4221,8 @@
   shows "uniformly_continuous_map m1 m2 f"
   unfolding uniformly_continuous_map_sequentially_alt imp_conjL
 proof (intro conjI strip)
-  show "f ` mspace m1 \<subseteq> mspace m2"
-    by (simp add: Cauchy_continuous_map_image f)
+  show "f \<in> mspace m1 \<rightarrow> mspace m2"
+    by (simp add: Cauchy_continuous_map_funspace f)
   interpret M1: Metric_space "mspace m1" "mdist m1"
     by (simp add: Metric_space_mspace_mdist)
   interpret M2: Metric_space "mspace m2" "mdist m2"
@@ -4286,7 +4286,7 @@
 lemma Lipschitz_continuous_map_projections:
   "Lipschitz_continuous_map (prod_metric m1 m2) m1 fst"
   "Lipschitz_continuous_map (prod_metric m1 m2) m2 snd"
-  by (simp add: Lipschitz_continuous_map_def prod_dist_def; 
+  by (simp add: Lipschitz_continuous_map_def prod_dist_def fst_Pi snd_Pi; 
       metis mult_numeral_1 real_sqrt_sum_squares_ge1 real_sqrt_sum_squares_ge2)+
 
 lemma Lipschitz_continuous_map_pairwise:
@@ -4308,11 +4308,11 @@
     show ?thesis
       unfolding Lipschitz_continuous_map_pos
     proof (intro exI conjI strip)
-      have f1im: "f1 ` mspace m \<subseteq> mspace m1"
+      have f1im: "f1 \<in> mspace m \<rightarrow> mspace m1"
         by (simp add: Lipschitz_continuous_map_image f1)
-      moreover have f2im: "f2 ` mspace m \<subseteq> mspace m2"
+      moreover have f2im: "f2 \<in> mspace m \<rightarrow> mspace m2"
         by (simp add: Lipschitz_continuous_map_image f2)
-      ultimately show "(\<lambda>x. (f1 x, f2 x)) ` mspace m \<subseteq> mspace (prod_metric m1 m2)"
+      ultimately show "(\<lambda>x. (f1 x, f2 x)) \<in> mspace m \<rightarrow> mspace (prod_metric m1 m2)"
         by auto
       show "B1+B2 > 0"
         using \<open>0 < B1\<close> \<open>0 < B2\<close> by linarith
@@ -4343,11 +4343,11 @@
     show ?thesis
       unfolding uniformly_continuous_map_def
     proof (intro conjI strip)
-      have f1im: "f1 ` mspace m \<subseteq> mspace m1"
-        by (simp add: uniformly_continuous_map_image f1)
-      moreover have f2im: "f2 ` mspace m \<subseteq> mspace m2"
-        by (simp add: uniformly_continuous_map_image f2)
-      ultimately show "(\<lambda>x. (f1 x, f2 x)) ` mspace m \<subseteq> mspace (prod_metric m1 m2)"
+      have f1im: "f1 \<in> mspace m \<rightarrow> mspace m1"
+        by (simp add: uniformly_continuous_map_funspace f1)
+      moreover have f2im: "f2 \<in> mspace m \<rightarrow> mspace m2"
+        by (simp add: uniformly_continuous_map_funspace f2)
+      ultimately show "(\<lambda>x. (f1 x, f2 x)) \<in> mspace m \<rightarrow> mspace (prod_metric m1 m2)"
         by auto
       fix \<epsilon>:: real
       assume "\<epsilon> > 0"
@@ -4402,9 +4402,9 @@
   assumes f: "Lipschitz_continuous_map m1 m2 f" and S: "Metric_space.mbounded (mspace m1) (mdist m1) S"
   shows "Metric_space.mbounded (mspace m2) (mdist m2) (f`S)"
 proof -
-  obtain B where fim: "f ` mspace m1 \<subseteq> mspace m2"
+  obtain B where fim: "f \<in> mspace m1 \<rightarrow> mspace m2"
     and "B>0" and B: "\<And>x y. \<lbrakk>x \<in> mspace m1; y \<in> mspace m1\<rbrakk> \<Longrightarrow> mdist m2 (f x) (f y) \<le> B * mdist m1 x y"
-    by (meson Lipschitz_continuous_map_pos f)
+    by (metis Lipschitz_continuous_map_pos f)
   show ?thesis
     unfolding Metric_space.mbounded_alt_pos [OF Metric_space_mspace_mdist]
   proof
@@ -4429,7 +4429,7 @@
   have "S \<subseteq> mspace m1"
     using S by (simp add: Metric_space.mtotally_bounded_sequentially[OF Metric_space_mspace_mdist])
   then show "f ` S \<subseteq> mspace m2"
-    using Cauchy_continuous_map_image f by blast
+    using Cauchy_continuous_map_funspace f by blast
   fix \<sigma> :: "nat \<Rightarrow> 'b"
   assume "range \<sigma> \<subseteq> f ` S"
   then have "\<forall>n. \<exists>x. \<sigma> n = f x \<and> x \<in> S"
@@ -4448,11 +4448,11 @@
 
 lemma Lipschitz_coefficient_pos:
   assumes "x \<in> mspace m" "y \<in> mspace m" "f x \<noteq> f y" 
-    and "f ` mspace m \<subseteq> mspace m2" 
+    and "f \<in> mspace m \<rightarrow> mspace m2" 
     and "\<And>x y. \<lbrakk>x \<in> mspace m; y \<in> mspace m\<rbrakk>
             \<Longrightarrow> mdist m2 (f x) (f y) \<le> k * mdist m x y"
   shows  "0 < k"
-  using assms by (smt (verit, best) image_subset_iff mdist_nonneg mdist_zero mult_nonpos_nonneg)
+  using assms by (smt (verit, best) Pi_iff mdist_nonneg mdist_zero mult_nonpos_nonneg)
 
 lemma Lipschitz_continuous_map_metric:
    "Lipschitz_continuous_map (prod_metric m m) euclidean_metric (\<lambda>(x,y). mdist m x y)"
@@ -4563,7 +4563,7 @@
 subsection \<open>Isometries\<close>
 
 lemma (in Metric_space12) isometry_imp_embedding_map:
-  assumes fim: "f ` M1 \<subseteq> M2" and d: "\<And>x y. \<lbrakk>x \<in> M1; y \<in> M1\<rbrakk> \<Longrightarrow> d2 (f x) (f y) = d1 x y"
+  assumes fim: "f \<in> M1 \<rightarrow> M2" and d: "\<And>x y. \<lbrakk>x \<in> M1; y \<in> M1\<rbrakk> \<Longrightarrow> d2 (f x) (f y) = d1 x y"
   shows "embedding_map M1.mtopology M2.mtopology f"
 proof -
   have "inj_on f M1"
@@ -4574,7 +4574,10 @@
     unfolding homeomorphic_maps_def
   proof (intro conjI; clarsimp)
     show "continuous_map M1.mtopology (subtopology M2.mtopology (f ` M1)) f"
-      by (metis M1.metric_continuous_map M1.topspace_mtopology M2.Metric_space_axioms continuous_map_into_subtopology d fim order_refl)
+    proof (rule continuous_map_into_subtopology)
+      show "continuous_map M1.mtopology M2.mtopology f"
+        by (metis M1.metric_continuous_map M2.Metric_space_axioms d fim image_subset_iff_funcset)
+    qed simp
     have "Lipschitz_continuous_map (submetric (metric(M2,d2)) (f ` M1)) (metric(M1,d1)) g"
       unfolding Lipschitz_continuous_map_def
     proof (intro conjI exI strip; simp)
@@ -4595,7 +4598,8 @@
 lemma (in Metric_space12) isometry_imp_homeomorphic_map:
   assumes fim: "f ` M1 = M2" and d: "\<And>x y. \<lbrakk>x \<in> M1; y \<in> M1\<rbrakk> \<Longrightarrow> d2 (f x) (f y) = d1 x y"
   shows "homeomorphic_map M1.mtopology M2.mtopology f"
-  by (metis M1.topspace_mtopology M2.subtopology_mspace d embedding_map_def fim isometry_imp_embedding_map subsetI)
+  by (metis image_eqI M1.topspace_mtopology M2.subtopology_mspace d embedding_map_def fim isometry_imp_embedding_map Pi_iff)
+
 subsection\<open>"Capped" equivalent bounded metrics and general product metrics\<close>
 
 definition (in Metric_space) capped_dist where
@@ -5132,6 +5136,5 @@
    "locally_connected_space(Euclidean_space n)"
   by (simp add: locally_path_connected_Euclidean_space locally_path_connected_imp_locally_connected_space)
 
-
 end
 
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Abstract_Topological_Spaces.thy
--- a/src/HOL/Analysis/Abstract_Topological_Spaces.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Abstract_Topological_Spaces.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -1819,13 +1819,13 @@
   qed
   moreover
   { assume R: ?rhs
-    then have fgim: "f ` topspace X \<subseteq> topspace X'" "g ` topspace Y \<subseteq> topspace Y'" 
+    then have fgim: "f \<in> topspace X \<rightarrow> topspace X'" "g \<in> topspace Y \<rightarrow> topspace Y'" 
           and cm: "closed_map X X' f" "closed_map Y Y' g"
       by (auto simp: proper_map_def closed_map_imp_subset_topspace)
     have "closed_map (prod_topology X Y) (prod_topology X' Y') h"
       unfolding closed_map_fibre_neighbourhood imp_conjL
     proof (intro conjI strip)
-      show "h ` topspace (prod_topology X Y) \<subseteq> topspace (prod_topology X' Y')"
+      show "h \<in> topspace (prod_topology X Y) \<rightarrow> topspace (prod_topology X' Y')"
         unfolding h_def using fgim by auto
       fix W w
       assume W: "openin (prod_topology X Y) W"
@@ -1960,7 +1960,7 @@
    "\<lbrakk>perfect_map X Z (g \<circ> f); f ` topspace X = topspace Y;
      continuous_map X Y f; continuous_map Y Z g; inj_on g (topspace Y)\<rbrakk>
     \<Longrightarrow> perfect_map X Y f"
-  by (meson continuous_map_image_subset_topspace perfect_map_def proper_map_from_composition_right_inj)
+  by (meson continuous_map_openin_preimage_eq perfect_map_def proper_map_from_composition_right_inj)
 
 
 subsection \<open>Regular spaces\<close>
@@ -2285,7 +2285,7 @@
   fix C assume "closedin Z C"
   then have "C \<subseteq> topspace Z"
     by (simp add: closedin_subset)
-  have "f ` topspace Z \<subseteq> topspace X" "g ` topspace Z \<subseteq> topspace Y"
+  have "f \<in> topspace Z \<rightarrow> topspace X" "g \<in> topspace Z \<rightarrow> topspace Y"
     by (simp_all add: assms closed_map_imp_subset_topspace)
   show "closedin (prod_topology X Y) ((\<lambda>x. (f x, g x)) ` C)"
     unfolding closedin_def topspace_prod_topology
@@ -2293,7 +2293,7 @@
     have "closedin Y (g ` C)"
       using \<open>closedin Z C\<close> assms(3) closed_map_def by blast
     with assms show "(\<lambda>x. (f x, g x)) ` C \<subseteq> topspace X \<times> topspace Y"
-      using \<open>C \<subseteq> topspace Z\<close> \<open>f ` topspace Z \<subseteq> topspace X\<close> continuous_map_closedin subsetD by fastforce
+      by (smt (verit) SigmaI \<open>closedin Z C\<close> closed_map_def closedin_subset image_subset_iff)
     have *: "\<exists>T. openin (prod_topology X Y) T \<and> (y1,y2) \<in> T \<and> T \<subseteq> topspace X \<times> topspace Y - (\<lambda>x. (f x, g x)) ` C"
       if "(y1,y2) \<notin> (\<lambda>x. (f x, g x)) ` C" and y1: "y1 \<in> topspace X" and y2: "y2 \<in> topspace Y"
       for y1 y2
@@ -5157,7 +5157,7 @@
 qed
 
 lemma closed_map_into_k_space:
-  assumes "k_space Y" and fim: "f ` (topspace X) \<subseteq> topspace Y"
+  assumes "k_space Y" and fim: "f \<in> (topspace X) \<rightarrow> topspace Y"
     and f: "\<And>K. compactin Y K
                 \<Longrightarrow> closed_map(subtopology X {x \<in> topspace X. f x \<in> K}) (subtopology Y K) f"
   shows "closed_map X Y f"
@@ -5176,13 +5176,13 @@
   qed
   then show "closedin Y (f ` C)"
     using \<open>k_space Y\<close> unfolding k_space
-    by (meson \<open>closedin X C\<close> closedin_subset dual_order.trans fim image_mono)
+    by (meson \<open>closedin X C\<close> closedin_subset order.trans fim funcset_image subset_image_iff)
 qed
 
 
 text \<open>Essentially the same proof\<close>
 lemma open_map_into_k_space:
-  assumes "k_space Y" and fim: "f ` (topspace X) \<subseteq> topspace Y"
+  assumes "k_space Y" and fim: "f \<in> (topspace X) \<rightarrow> topspace Y"
     and f: "\<And>K. compactin Y K
                  \<Longrightarrow> open_map (subtopology X {x \<in> topspace X. f x \<in> K}) (subtopology Y K) f"
   shows "open_map X Y f"
@@ -5201,7 +5201,7 @@
   qed
   then show "openin Y (f ` C)"
     using \<open>k_space Y\<close> unfolding k_space_open
-    by (meson \<open>openin X C\<close> openin_subset dual_order.trans fim image_mono)
+    by (meson \<open>openin X C\<close> openin_subset order.trans fim funcset_image subset_image_iff)
 qed
 
 lemma quotient_map_into_k_space:
@@ -5255,7 +5255,7 @@
 lemma open_map_into_k_space_eq:
   assumes "k_space Y"
   shows "open_map X Y f \<longleftrightarrow>
-         f ` (topspace X) \<subseteq> topspace Y \<and>
+         f \<in> (topspace X) \<rightarrow> topspace Y \<and>
          (\<forall>k. compactin Y k
               \<longrightarrow> open_map (subtopology X {x \<in> topspace X. f x \<in> k}) (subtopology Y k) f)"
        (is "?lhs=?rhs")
@@ -5269,7 +5269,7 @@
 lemma closed_map_into_k_space_eq:
   assumes "k_space Y"
   shows "closed_map X Y f \<longleftrightarrow>
-         f ` (topspace X) \<subseteq> topspace Y \<and>
+         f \<in> (topspace X) \<rightarrow> topspace Y \<and>
          (\<forall>k. compactin Y k
               \<longrightarrow> closed_map (subtopology X {x \<in> topspace X. f x \<in> k}) (subtopology Y k) f)"
        (is "?lhs \<longleftrightarrow> ?rhs")
@@ -5281,7 +5281,7 @@
 qed
 
 lemma proper_map_into_k_space:
-  assumes "k_space Y" and fim: "f ` (topspace X) \<subseteq> topspace Y"
+  assumes "k_space Y" and fim: "f \<in> (topspace X) \<rightarrow> topspace Y"
     and f: "\<And>K. compactin Y K
                  \<Longrightarrow> proper_map (subtopology X {x \<in> topspace X. f x \<in> K})
                                 (subtopology Y K) f"
@@ -5297,7 +5297,7 @@
 lemma proper_map_into_k_space_eq:
   assumes "k_space Y"
   shows "proper_map X Y f \<longleftrightarrow>
-         f ` (topspace X) \<subseteq> topspace Y \<and>
+         f \<in> (topspace X) \<rightarrow> topspace Y \<and>
          (\<forall>K. compactin Y K
               \<longrightarrow> proper_map (subtopology X {x \<in> topspace X. f x \<in> K}) (subtopology Y K) f)"
          (is "?lhs \<longleftrightarrow> ?rhs")
@@ -5305,11 +5305,11 @@
   show "?lhs \<Longrightarrow> ?rhs"
     by (simp add: proper_map_imp_subset_topspace proper_map_restriction)
   show "?rhs \<Longrightarrow> ?lhs"
-    by (simp add: assms proper_map_into_k_space)
+    by (simp add: assms funcset_image proper_map_into_k_space)
 qed
 
 lemma compact_imp_proper_map:
-  assumes "k_space Y" "kc_space Y" and fim: "f ` (topspace X) \<subseteq> topspace Y" 
+  assumes "k_space Y" "kc_space Y" and fim: "f \<in> (topspace X) \<rightarrow> topspace Y" 
     and f: "continuous_map X Y f \<or> kc_space X" 
     and comp: "\<And>K. compactin Y K \<Longrightarrow> compactin X {x \<in> topspace X. f x \<in> K}"
   shows "proper_map X Y f"
@@ -5325,12 +5325,12 @@
   assumes "k_space Y" "kc_space Y" 
     and f: "continuous_map X Y f \<or> kc_space X" 
   shows  "proper_map X Y f \<longleftrightarrow>
-             f ` (topspace X) \<subseteq> topspace Y \<and>
+             f \<in> (topspace X) \<rightarrow> topspace Y \<and>
              (\<forall>K. compactin Y K \<longrightarrow> compactin X {x \<in> topspace X. f x \<in> K})"
          (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   show "?lhs \<Longrightarrow> ?rhs"
-    by (simp add: proper_map_alt proper_map_imp_subset_topspace)
+    using \<open>k_space Y\<close> compactin_proper_map_preimage proper_map_into_k_space_eq by blast
 qed (use assms compact_imp_proper_map in auto)
 
 lemma compact_imp_perfect_map:
@@ -5338,7 +5338,7 @@
     and "continuous_map X Y f" 
     and "\<And>K. compactin Y K \<Longrightarrow> compactin X {x \<in> topspace X. f x \<in> K}"
   shows "perfect_map X Y f"
-  by (simp add: assms compact_imp_proper_map perfect_map_def)
+  by (simp add: assms compact_imp_proper_map perfect_map_def flip: image_subset_iff_funcset)
 
 end
 
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Abstract_Topology.thy
--- a/src/HOL/Analysis/Abstract_Topology.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Abstract_Topology.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -1703,24 +1703,24 @@
 
 
 lemma continuous_map_from_subtopology:
-     "continuous_map X X' f \<Longrightarrow> continuous_map (subtopology X S) X' f"
+     "continuous_map X Y f \<Longrightarrow> continuous_map (subtopology X S) Y f"
   by (auto simp: continuous_map openin_subtopology)
 
 lemma continuous_map_into_fulltopology:
-   "continuous_map X (subtopology X' T) f \<Longrightarrow> continuous_map X X' f"
+   "continuous_map X (subtopology Y T) f \<Longrightarrow> continuous_map X Y f"
   by (auto simp: continuous_map_in_subtopology)
 
 lemma continuous_map_into_subtopology:
-   "\<lbrakk>continuous_map X X' f; f ` topspace X \<subseteq> T\<rbrakk> \<Longrightarrow> continuous_map X (subtopology X' T) f"
+   "\<lbrakk>continuous_map X Y f; f \<in> topspace X \<rightarrow> T\<rbrakk> \<Longrightarrow> continuous_map X (subtopology Y T) f"
   by (auto simp: continuous_map_in_subtopology)
 
 lemma continuous_map_from_subtopology_mono:
-     "\<lbrakk>continuous_map (subtopology X T) X' f; S \<subseteq> T\<rbrakk>
-      \<Longrightarrow> continuous_map (subtopology X S) X' f"
+     "\<lbrakk>continuous_map (subtopology X T) Y f; S \<subseteq> T\<rbrakk>
+      \<Longrightarrow> continuous_map (subtopology X S) Y f"
   by (metis inf.absorb_iff2 continuous_map_from_subtopology subtopology_subtopology)
 
 lemma continuous_map_from_discrete_topology [simp]:
-  "continuous_map (discrete_topology U) X f \<longleftrightarrow> f ` U \<subseteq> topspace X"
+  "continuous_map (discrete_topology U) X f \<longleftrightarrow> f \<in> U \<rightarrow> topspace X"
   by (auto simp: continuous_map_def)
 
 lemma continuous_map_iff_continuous [simp]: "continuous_map (top_of_set S) euclidean g = continuous_on S g"
@@ -1731,12 +1731,12 @@
 
 lemma continuous_map_openin_preimage_eq:
    "continuous_map X Y f \<longleftrightarrow>
-        f ` (topspace X) \<subseteq> topspace Y \<and> (\<forall>U. openin Y U \<longrightarrow> openin X (topspace X \<inter> f -` U))"
+        f \<in> (topspace X) \<rightarrow> topspace Y \<and> (\<forall>U. openin Y U \<longrightarrow> openin X (topspace X \<inter> f -` U))"
   by (auto simp: continuous_map_def vimage_def Int_def)
 
 lemma continuous_map_closedin_preimage_eq:
    "continuous_map X Y f \<longleftrightarrow>
-        f ` (topspace X) \<subseteq> topspace Y \<and> (\<forall>U. closedin Y U \<longrightarrow> closedin X (topspace X \<inter> f -` U))"
+        f \<in> (topspace X) \<rightarrow> topspace Y \<and> (\<forall>U. closedin Y U \<longrightarrow> closedin X (topspace X \<inter> f -` U))"
   by (auto simp: continuous_map_closedin vimage_def Int_def)
 
 lemma continuous_map_square_root: "continuous_map euclideanreal euclideanreal sqrt"
@@ -1759,7 +1759,7 @@
 
 lemma\<^marker>\<open>tag important\<close> continuous_map_alt:
    "continuous_map T1 T2 f 
-    = ((\<forall>U. openin T2 U \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)) \<and> f ` topspace T1 \<subseteq> topspace T2)"
+    = ((\<forall>U. openin T2 U \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)) \<and> f \<in> topspace T1 \<rightarrow> topspace T2)"
   by (auto simp: continuous_map_def vimage_def image_def Collect_conj_eq inf_commute)
 
 lemma continuous_map_open [intro]:
@@ -1782,8 +1782,8 @@
   where "closed_map X1 X2 f \<equiv> \<forall>U. closedin X1 U \<longrightarrow> closedin X2 (f ` U)"
 
 lemma open_map_imp_subset_topspace:
-     "open_map X1 X2 f \<Longrightarrow> f ` (topspace X1) \<subseteq> topspace X2"
-  unfolding open_map_def by (simp add: openin_subset)
+     "open_map X1 X2 f \<Longrightarrow> f \<in> (topspace X1) \<rightarrow> topspace X2"
+  unfolding open_map_def using openin_subset by fastforce
 
 lemma open_map_on_empty:
    "topspace X = {} \<Longrightarrow> open_map X Y f"
@@ -1798,8 +1798,8 @@
   by (metis closed_map_def closed_map_on_empty closedin_empty closedin_topspace image_constant_conv)
 
 lemma open_map_imp_subset:
-    "\<lbrakk>open_map X1 X2 f; S \<subseteq> topspace X1\<rbrakk> \<Longrightarrow> f ` S \<subseteq> topspace X2"
-  by (meson order_trans open_map_imp_subset_topspace subset_image_iff)
+    "\<lbrakk>open_map X1 X2 f; S \<subseteq> topspace X1\<rbrakk> \<Longrightarrow> f \<in> S \<rightarrow> topspace X2"
+  using open_map_imp_subset_topspace by fastforce
 
 lemma topology_finer_open_id:
      "(\<forall>S. openin X S \<longrightarrow> openin X' S) \<longleftrightarrow> open_map X X' id"
@@ -1826,11 +1826,11 @@
   by (metis (no_types, lifting) image_comp open_map_def)
 
 lemma closed_map_imp_subset_topspace:
-     "closed_map X1 X2 f \<Longrightarrow> f ` (topspace X1) \<subseteq> topspace X2"
-  by (simp add: closed_map_def closedin_subset)
+     "closed_map X1 X2 f \<Longrightarrow> f \<in> (topspace X1) \<rightarrow> topspace X2"
+  by (simp add: closed_map_def closedin_def image_subset_iff_funcset)
 
 lemma closed_map_imp_subset:
-     "\<lbrakk>closed_map X1 X2 f; S \<subseteq> topspace X1\<rbrakk> \<Longrightarrow> f ` S \<subseteq> topspace X2"
+     "\<lbrakk>closed_map X1 X2 f; S \<subseteq> topspace X1\<rbrakk> \<Longrightarrow> f \<in> S \<rightarrow> topspace X2"
   using closed_map_imp_subset_topspace by blast
 
 lemma topology_finer_closed_id:
@@ -1863,21 +1863,21 @@
   by (simp add: closed_map_inclusion_eq closedin_Int)
 
 lemma open_map_into_subtopology:
-    "\<lbrakk>open_map X X' f; f ` topspace X \<subseteq> S\<rbrakk> \<Longrightarrow> open_map X (subtopology X' S) f"
+    "\<lbrakk>open_map X X' f; f \<in> topspace X \<rightarrow> S\<rbrakk> \<Longrightarrow> open_map X (subtopology X' S) f"
   unfolding open_map_def openin_subtopology
   using openin_subset by fastforce
 
 lemma closed_map_into_subtopology:
-    "\<lbrakk>closed_map X X' f; f ` topspace X \<subseteq> S\<rbrakk> \<Longrightarrow> closed_map X (subtopology X' S) f"
+    "\<lbrakk>closed_map X X' f; f \<in> topspace X \<rightarrow> S\<rbrakk> \<Longrightarrow> closed_map X (subtopology X' S) f"
   unfolding closed_map_def closedin_subtopology
   using closedin_subset by fastforce
 
 lemma open_map_into_discrete_topology:
-    "open_map X (discrete_topology U) f \<longleftrightarrow> f ` (topspace X) \<subseteq> U"
+    "open_map X (discrete_topology U) f \<longleftrightarrow> f \<in> (topspace X) \<rightarrow> U"
   unfolding open_map_def openin_discrete_topology using openin_subset by blast
 
 lemma closed_map_into_discrete_topology:
-    "closed_map X (discrete_topology U) f \<longleftrightarrow> f ` (topspace X) \<subseteq> U"
+    "closed_map X (discrete_topology U) f \<longleftrightarrow> f \<in> (topspace X) \<rightarrow> U"
   unfolding closed_map_def closedin_discrete_topology using closedin_subset by blast
 
 lemma bijective_open_imp_closed_map:
@@ -1930,8 +1930,8 @@
 
 lemma closed_map_closure_of_image:
    "closed_map X Y f \<longleftrightarrow>
-        f ` topspace X \<subseteq> topspace Y \<and>
-        (\<forall>S. S \<subseteq> topspace X \<longrightarrow> Y closure_of (f ` S) \<subseteq> image f (X closure_of S))" (is "?lhs=?rhs")
+        f \<in> topspace X \<rightarrow> topspace Y \<and>
+        (\<forall>S. S \<subseteq> topspace X \<longrightarrow> Y closure_of (f ` S) \<subseteq> f ` (X closure_of S))" (is "?lhs=?rhs")
 proof
   assume ?lhs
   then show ?rhs
@@ -1953,12 +1953,12 @@
 
 lemma open_map_interior_of_image_subset_gen:
   "open_map X Y f \<longleftrightarrow>
-       (f ` topspace X \<subseteq> topspace Y \<and> (\<forall>S. f ` (X interior_of S) \<subseteq> Y interior_of f ` S))"
-  by (meson open_map_imp_subset_topspace open_map_interior_of_image_subset)
+       (f \<in> topspace X \<rightarrow> topspace Y \<and> (\<forall>S. f ` (X interior_of S) \<subseteq> Y interior_of f ` S))"
+  by (metis open_map_imp_subset_topspace open_map_interior_of_image_subset)
 
 lemma open_map_preimage_neighbourhood:
    "open_map X Y f \<longleftrightarrow>
-    (f ` topspace X \<subseteq> topspace Y \<and>
+    (f \<in> topspace X \<rightarrow> topspace Y \<and>
      (\<forall>U T. closedin X U \<and> T \<subseteq> topspace Y \<and>
             {x \<in> topspace X. f x \<in> T} \<subseteq> U \<longrightarrow>
             (\<exists>V. closedin Y V \<and> T \<subseteq> V \<and> {x \<in> topspace X. f x \<in> V} \<subseteq> U)))" (is "?lhs=?rhs")
@@ -1966,7 +1966,7 @@
   assume L: ?lhs
   show ?rhs
   proof (intro conjI strip)
-    show "f ` topspace X \<subseteq> topspace Y"
+    show "f \<in> topspace X \<rightarrow> topspace Y"
       by (simp add: \<open>open_map X Y f\<close> open_map_imp_subset_topspace)
   next
     fix U T
@@ -2000,7 +2000,7 @@
 
 lemma closed_map_preimage_neighbourhood:
    "closed_map X Y f \<longleftrightarrow>
-        image f (topspace X) \<subseteq> topspace Y \<and>
+        f \<in> topspace X \<rightarrow> topspace Y \<and>
         (\<forall>U T. openin X U \<and> T \<subseteq> topspace Y \<and>
               {x \<in> topspace X. f x \<in> T} \<subseteq> U
               \<longrightarrow> (\<exists>V. openin Y V \<and> T \<subseteq> V \<and>
@@ -2009,7 +2009,7 @@
   assume L: ?lhs
   show ?rhs
   proof (intro conjI strip)
-    show "f ` topspace X \<subseteq> topspace Y"
+    show "f \<in> topspace X \<rightarrow> topspace Y"
       by (simp add: L closed_map_imp_subset_topspace)
   next
     fix U T
@@ -2042,13 +2042,13 @@
 
 lemma closed_map_fibre_neighbourhood:
   "closed_map X Y f \<longleftrightarrow>
-     f ` (topspace X) \<subseteq> topspace Y \<and>
+     f \<in> (topspace X) \<rightarrow> topspace Y \<and>
      (\<forall>U y. openin X U \<and> y \<in> topspace Y \<and> {x \<in> topspace X. f x = y} \<subseteq> U
      \<longrightarrow> (\<exists>V. openin Y V \<and> y \<in> V \<and> {x \<in> topspace X. f x \<in> V} \<subseteq> U))"
   unfolding closed_map_preimage_neighbourhood
 proof (intro conj_cong refl all_cong1)
   fix U
-  assume "f ` topspace X \<subseteq> topspace Y"
+  assume "f \<in> topspace X \<rightarrow> topspace Y"
   show "(\<forall>T. openin X U \<and> T \<subseteq> topspace Y \<and> {x \<in> topspace X. f x \<in> T} \<subseteq> U 
          \<longrightarrow> (\<exists>V. openin Y V \<and> T \<subseteq> V \<and> {x \<in> topspace X. f x \<in> V} \<subseteq> U))
       = (\<forall>y. openin X U \<and> y \<in> topspace Y \<and> {x \<in> topspace X. f x = y} \<subseteq> U 
@@ -2083,8 +2083,8 @@
 
 lemma open_map_in_subtopology:
    "openin Y S
-        \<Longrightarrow> (open_map X (subtopology Y S) f \<longleftrightarrow> open_map X Y f \<and> f ` (topspace X) \<subseteq> S)"
-  by (metis le_inf_iff open_map_def open_map_imp_subset_topspace open_map_into_subtopology openin_trans_full topspace_subtopology)
+        \<Longrightarrow> open_map X (subtopology Y S) f \<longleftrightarrow> open_map X Y f \<and> f \<in> topspace X \<rightarrow> S"
+  by (metis image_subset_iff_funcset open_map_def open_map_into_subtopology openin_imp_subset openin_topspace openin_trans_full)
 
 lemma open_map_from_open_subtopology:
    "\<lbrakk>openin Y S; open_map X (subtopology Y S) f\<rbrakk> \<Longrightarrow> open_map X Y f"
@@ -2092,7 +2092,7 @@
 
 lemma closed_map_in_subtopology:
    "closedin Y S
-        \<Longrightarrow> closed_map X (subtopology Y S) f \<longleftrightarrow> (closed_map X Y f \<and> f ` topspace X \<subseteq> S)"
+        \<Longrightarrow> closed_map X (subtopology Y S) f \<longleftrightarrow> (closed_map X Y f \<and> f \<in> topspace X \<rightarrow> S)"
   by (metis closed_map_def closed_map_imp_subset_topspace closed_map_into_subtopology 
         closedin_closed_subtopology closedin_subset topspace_subtopology_subset)
 
@@ -2138,7 +2138,7 @@
 qed
 
 lemma closed_map_from_composition_right:
-  assumes cmf: "closed_map X Z (g \<circ> f)" "f ` topspace X \<subseteq> topspace Y" "continuous_map Y Z g" "inj_on g (topspace Y)"
+  assumes cmf: "closed_map X Z (g \<circ> f)" "f \<in> topspace X \<rightarrow> topspace Y" "continuous_map Y Z g" "inj_on g (topspace Y)"
   shows "closed_map X Y f"
   unfolding closed_map_def
 proof (intro strip)
@@ -2156,7 +2156,7 @@
 
 text \<open>identical proof as the above\<close>
 lemma open_map_from_composition_right:
-  assumes "open_map X Z (g \<circ> f)" "f ` topspace X \<subseteq> topspace Y" "continuous_map Y Z g" "inj_on g (topspace Y)"
+  assumes "open_map X Z (g \<circ> f)" "f \<in> topspace X \<rightarrow> topspace Y" "continuous_map Y Z g" "inj_on g (topspace Y)"
   shows "open_map X Y f"
   unfolding open_map_def
 proof (intro strip)
@@ -2733,8 +2733,8 @@
   by (simp add: homeomorphic_eq_everything_map)
 
 lemma homeomorphic_imp_surjective_map:
-   "homeomorphic_map X Y f \<Longrightarrow> f ` (topspace X) = topspace Y"
-  by (simp add: homeomorphic_eq_everything_map)
+   "homeomorphic_map X Y f \<Longrightarrow> f ` topspace X = topspace Y"
+  using homeomorphic_eq_everything_map by fastforce
 
 lemma homeomorphic_imp_injective_map:
     "homeomorphic_map X Y f \<Longrightarrow> inj_on f (topspace X)"
@@ -2899,13 +2899,13 @@
 
 lemma all_openin_homeomorphic_image:
   assumes "homeomorphic_map X Y f"
-  shows "(\<forall>V. openin Y V \<longrightarrow> P V) \<longleftrightarrow> (\<forall>U. openin X U \<longrightarrow> P(f ` U))" 
-  by (metis (no_types, lifting) assms homeomorphic_imp_surjective_map homeomorphic_map_openness openin_subset subset_image_iff)
+  shows "(\<forall>V. openin Y V \<longrightarrow> P V) \<longleftrightarrow> (\<forall>U. openin X U \<longrightarrow> P(f ` U))"
+  by (metis (no_types, lifting) assms homeomorphic_eq_everything_map homeomorphic_map_openness openin_subset subset_image_iff) 
 
 lemma all_closedin_homeomorphic_image:
   assumes "homeomorphic_map X Y f"
-  shows "(\<forall>V. closedin Y V \<longrightarrow> P V) \<longleftrightarrow> (\<forall>U. closedin X U \<longrightarrow> P(f ` U))"  (is "?lhs = ?rhs")
-  by (metis (no_types, lifting) assms homeomorphic_imp_surjective_map homeomorphic_map_closedness closedin_subset subset_image_iff)
+  shows "(\<forall>V. closedin Y V \<longrightarrow> P V) \<longleftrightarrow> (\<forall>U. closedin X U \<longrightarrow> P(f ` U))"
+  by (metis (no_types, lifting) assms closedin_subset homeomorphic_eq_everything_map homeomorphic_map_closedness subset_image_iff)  
 
 
 lemma homeomorphic_map_derived_set_of:
@@ -3059,7 +3059,7 @@
 
 lemma homeomorphic_empty_space:
      "X homeomorphic_space Y \<Longrightarrow> topspace X = {} \<longleftrightarrow> topspace Y = {}"
-  by (metis homeomorphic_imp_surjective_map homeomorphic_space image_is_empty)
+  by (metis homeomorphic_eq_everything_map homeomorphic_space image_is_empty)
 
 lemma homeomorphic_empty_space_eq:
   assumes "topspace X = {}"
@@ -3067,6 +3067,14 @@
   unfolding homeomorphic_maps_def homeomorphic_space_def
   by (metis assms continuous_map_on_empty continuous_map_closedin ex_in_conv)
 
+lemma homeomorphic_space_unfold:
+  assumes "X homeomorphic_space Y"
+  obtains f g where  "homeomorphic_map X Y f"  "homeomorphic_map Y X g"
+    and "\<And>x. x \<in> topspace X \<Longrightarrow> g(f x) = x" "\<And>y. y \<in> topspace Y \<Longrightarrow> f(g y) = y"
+    and "f \<in> topspace X \<rightarrow> topspace Y"  "g \<in> topspace Y \<rightarrow> topspace X"
+  using assms unfolding homeomorphic_space_def homeomorphic_maps_map
+  by (smt (verit, best) Pi_I homeomorphic_imp_surjective_map image_eqI)
+
 subsection\<open>Connected topological spaces\<close>
 
 definition connected_space :: "'a topology \<Rightarrow> bool" where
@@ -3923,9 +3931,9 @@
   have hm: "homeomorphic_map X (subtopology X' (f ` topspace X)) f" "homeomorphic_map X' (subtopology X'' (g ` topspace X')) g"
     using assms by (auto simp: embedding_map_def)
   then obtain C where "g ` topspace X' \<inter> C = (g \<circ> f) ` topspace X"
-    by (metis homeomorphic_imp_surjective_map image_comp image_mono inf.absorb_iff2 topspace_subtopology)
+    using continuous_map_image_subset_topspace homeomorphic_imp_continuous_map by fastforce
   then have "homeomorphic_map (subtopology X' (f ` topspace X)) (subtopology X'' ((g \<circ> f) ` topspace X)) g"
-    by (metis hm homeomorphic_imp_surjective_map homeomorphic_map_subtopologies image_comp subtopology_subtopology topspace_subtopology)
+    by (metis hm homeomorphic_eq_everything_map homeomorphic_map_subtopologies image_comp subtopology_subtopology topspace_subtopology)
   then show ?thesis
   unfolding embedding_map_def
   using hm(1) homeomorphic_map_compose by blast
@@ -3945,14 +3953,15 @@
 
 lemma injective_open_imp_embedding_map:
    "\<lbrakk>continuous_map X Y f; open_map X Y f; inj_on f (topspace X)\<rbrakk> \<Longrightarrow> embedding_map X Y f"
-  unfolding embedding_map_def
-  by (simp add: continuous_map_in_subtopology continuous_open_quotient_map eq_iff homeomorphic_map_def open_map_imp_subset open_map_into_subtopology)
+  unfolding embedding_map_def homeomorphic_map_def
+  by (simp add: image_subset_iff_funcset continuous_map_in_subtopology continuous_open_quotient_map eq_iff
+      open_map_imp_subset open_map_into_subtopology)
 
 lemma injective_closed_imp_embedding_map:
   "\<lbrakk>continuous_map X Y f; closed_map X Y f; inj_on f (topspace X)\<rbrakk> \<Longrightarrow> embedding_map X Y f"
-  unfolding embedding_map_def
+  unfolding embedding_map_def homeomorphic_map_def
   by (simp add: closed_map_imp_subset closed_map_into_subtopology continuous_closed_quotient_map 
-                continuous_map_in_subtopology dual_order.eq_iff homeomorphic_map_def)
+                continuous_map_in_subtopology dual_order.eq_iff image_subset_iff_funcset)
 
 lemma embedding_map_imp_homeomorphic_space:
    "embedding_map X Y f \<Longrightarrow> X homeomorphic_space (subtopology Y (f ` (topspace X)))"
@@ -4098,7 +4107,7 @@
 lemma continuous_map_euclidean_top_of_set:
   assumes eq: "f -` S = UNIV" and cont: "continuous_on UNIV f"
   shows "continuous_map euclidean (top_of_set S) f"
-  by (simp add: cont continuous_map_into_subtopology eq image_subset_iff_subset_vimage)
+  by (simp add: cont continuous_map_in_subtopology eq image_subset_iff_subset_vimage)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Half-global and completely global cases\<close>
@@ -4256,36 +4265,32 @@
 qed
 
 lemma continuous_on_open_gen:
-  assumes "f ` S \<subseteq> T"
+  assumes "f \<in> S \<rightarrow> T"
     shows "continuous_on S f \<longleftrightarrow>
              (\<forall>U. openin (top_of_set T) U
                   \<longrightarrow> openin (top_of_set S) (S \<inter> f -` U))"
      (is "?lhs = ?rhs")
 proof
   assume ?lhs
-  then show ?rhs
-    by (clarsimp simp add: continuous_openin_preimage_eq openin_open)
-      (metis Int_assoc assms image_subset_iff_subset_vimage inf.absorb_iff1)
+  with assms show ?rhs
+    apply (simp add: Pi_iff continuous_openin_preimage_eq openin_open)
+    by (metis inf.orderE inf_assoc subsetI vimageI vimage_Int)
 next
   assume R [rule_format]: ?rhs
   show ?lhs
   proof (clarsimp simp add: continuous_openin_preimage_eq)
-    fix U::"'a set"
-    assume "open U"
-    then have "openin (top_of_set S) (S \<inter> f -` (U \<inter> T))"
-      by (metis R inf_commute openin_open)
-    then show "openin (top_of_set S) (S \<inter> f -` U)"
-      by (metis Int_assoc Int_commute assms image_subset_iff_subset_vimage inf.absorb_iff2 vimage_Int)
+    show "\<And>T. open T \<Longrightarrow> openin (top_of_set S) (S \<inter> f -` T)"
+      by (metis (no_types) R assms image_subset_iff_funcset image_subset_iff_subset_vimage inf.orderE inf_assoc openin_open_Int vimage_Int)
   qed
 qed
 
 lemma continuous_openin_preimage:
-  "\<lbrakk>continuous_on S f; f ` S \<subseteq> T; openin (top_of_set T) U\<rbrakk>
+  "\<lbrakk>continuous_on S f; f \<in> S \<rightarrow> T; openin (top_of_set T) U\<rbrakk>
         \<Longrightarrow> openin (top_of_set S) (S \<inter> f -` U)"
   by (simp add: continuous_on_open_gen)
 
 lemma continuous_on_closed_gen:
-  assumes "f ` S \<subseteq> T"
+  assumes "f \<in> S \<rightarrow> T"
   shows "continuous_on S f \<longleftrightarrow>
              (\<forall>U. closedin (top_of_set T) U
                   \<longrightarrow> closedin (top_of_set S) (S \<inter> f -` U))"
@@ -4298,7 +4303,7 @@
 qed
 
 lemma continuous_closedin_preimage_gen:
-  assumes "continuous_on S f" "f ` S \<subseteq> T" "closedin (top_of_set T) U"
+  assumes "continuous_on S f" "f \<in> S \<rightarrow> T" "closedin (top_of_set T) U"
     shows "closedin (top_of_set S) (S \<inter> f -` U)"
 using assms continuous_on_closed_gen by blast
 
@@ -4825,7 +4830,7 @@
   by (simp add: proper_map_def)
 
 lemma proper_map_imp_subset_topspace:
-   "proper_map X Y f \<Longrightarrow> f ` (topspace X) \<subseteq> topspace Y"
+   "proper_map X Y f \<Longrightarrow> f \<in> (topspace X) \<rightarrow> topspace Y"
   by (simp add: closed_map_imp_subset_topspace proper_map_def)
 
 lemma proper_map_restriction:
@@ -4868,7 +4873,7 @@
   assumes f: "proper_map X Y f" and "compactin Y K"
   shows "compactin X {x. x \<in> topspace X \<and> f x \<in> K}"
 proof -
-  have "f ` (topspace X) \<subseteq> topspace Y"
+  have "f \<in> (topspace X) \<rightarrow> topspace Y"
     by (simp add: f proper_map_imp_subset_topspace)
   have *: "\<And>y. y \<in> topspace Y \<Longrightarrow> compactin X {x \<in> topspace X. f x = y}"
     using f by (auto simp: proper_map_def)
@@ -5009,7 +5014,7 @@
   by (metis closed_Int_compactin closed_map_inclusion_eq inf.absorb_iff2 inj_on_id injective_imp_proper_eq_closed_map)
 
 lemma proper_map_into_subtopology:
-   "\<lbrakk>proper_map X Y f; f ` topspace X \<subseteq> C\<rbrakk> \<Longrightarrow> proper_map X (subtopology Y C) f"
+   "\<lbrakk>proper_map X Y f; f \<in> topspace X \<rightarrow> C\<rbrakk> \<Longrightarrow> proper_map X (subtopology Y C) f"
   by (simp add: closed_map_into_subtopology compactin_subtopology proper_map_alt)
 
 lemma proper_map_from_composition_left:
@@ -5031,7 +5036,7 @@
 qed
 
 lemma proper_map_from_composition_right_inj:
-  assumes gf: "proper_map X Z (g \<circ> f)" and fim: "f ` topspace X \<subseteq> topspace Y" 
+  assumes gf: "proper_map X Z (g \<circ> f)" and fim: "f \<in> topspace X \<rightarrow> topspace Y" 
     and contf: "continuous_map Y Z g" and inj: "inj_on g (topspace Y)"
   shows "proper_map X Y f"
   unfolding proper_map_def
@@ -5041,7 +5046,7 @@
   fix y
   assume "y \<in> topspace Y"
   with fim inj have eq: "{x \<in> topspace X. f x = y} = {x \<in> topspace X. (g \<circ> f) x = g y}"
-    by (auto simp: image_subset_iff inj_onD)
+    by (auto simp: Pi_iff inj_onD)
   show "compactin X {x \<in> topspace X. f x = y}"
     unfolding eq
     by (smt (verit) Collect_cong \<open>y \<in> topspace Y\<close> contf continuous_map_closedin gf proper_map_def)
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Abstract_Topology_2.thy
--- a/src/HOL/Analysis/Abstract_Topology_2.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Abstract_Topology_2.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -575,7 +575,7 @@
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Quotient maps\<close>
 
 lemma quotient_map_imp_continuous_open:
-  assumes T: "f ` S \<subseteq> T"
+  assumes T: "f \<in> S \<rightarrow> T"
       and ope: "\<And>U. U \<subseteq> T
               \<Longrightarrow> (openin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
                    openin (top_of_set T) U)"
@@ -587,7 +587,7 @@
 qed
 
 lemma quotient_map_imp_continuous_closed:
-  assumes T: "f ` S \<subseteq> T"
+  assumes T: "f \<in> S \<rightarrow> T"
       and ope: "\<And>U. U \<subseteq> T
                   \<Longrightarrow> (closedin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
                        closedin (top_of_set T) U)"
@@ -635,8 +635,8 @@
 qed
 
 lemma continuous_right_inverse_imp_quotient_map:
-  assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T"
-      and contg: "continuous_on T g" and img: "g ` T \<subseteq> S"
+  assumes contf: "continuous_on S f" and imf: "f \<in> S \<rightarrow> T"
+      and contg: "continuous_on T g" and img: "g \<in> T \<rightarrow> S"
       and fg [simp]: "\<And>y. y \<in> T \<Longrightarrow> f(g y) = y"
       and U: "U \<subseteq> T"
     shows "openin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
@@ -657,13 +657,13 @@
     finally have eq: "T \<inter> g -` (g ` T \<inter> (S \<inter> f -` U)) = U" .
     assume ?lhs
     then have *: "openin (top_of_set (g ` T)) (g ` T \<inter> (S \<inter> f -` U))"
-      by (meson img openin_Int openin_subtopology_Int_subset openin_subtopology_self)
+      by (meson img openin_Int openin_subtopology_Int_subset openin_subtopology_self image_subset_iff_funcset)
     show ?rhs
       using g [OF *] eq by auto
   next
     assume rhs: ?rhs
     show ?lhs
-      by (metis f fg image_eqI image_subset_iff imf img openin_subopen openin_subtopology_self openin_trans rhs)
+      using assms continuous_openin_preimage rhs by blast
   qed
 qed
 
@@ -696,7 +696,7 @@
     shows "continuous_map X Y g"
   unfolding continuous_map_openin_preimage_eq
 proof (intro conjI allI impI)
-  show "g ` topspace X \<subseteq> topspace Y"
+  show "g \<in> topspace X \<rightarrow> topspace Y"
     using g cont continuous_map_image_subset_topspace by fastforce
 next
   fix U
@@ -737,7 +737,7 @@
   shows "continuous_map X Y g"
   unfolding continuous_map_closedin_preimage_eq
 proof (intro conjI allI impI)
-  show "g ` topspace X \<subseteq> topspace Y"
+  show "g \<in> topspace X \<rightarrow> topspace Y"
     using g cont continuous_map_image_subset_topspace by fastforce
 next
   fix U
@@ -885,7 +885,7 @@
 
 definition\<^marker>\<open>tag important\<close> retraction :: "('a::topological_space) set \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool"
 where "retraction S T r \<longleftrightarrow>
-  T \<subseteq> S \<and> continuous_on S r \<and> r ` S \<subseteq> T \<and> (\<forall>x\<in>T. r x = x)"
+  T \<subseteq> S \<and> continuous_on S r \<and> r \<in> S \<rightarrow> T \<and> (\<forall>x\<in>T. r x = x)"
 
 definition\<^marker>\<open>tag important\<close> retract_of (infixl "retract'_of" 50) where
 "T retract_of S  \<longleftrightarrow>  (\<exists>r. retraction S T r)"
@@ -899,20 +899,20 @@
   fixes S :: "'a::topological_space set"
     and T :: "'b::topological_space set"
   assumes contt: "continuous_on T i"
-    and "i ` T \<subseteq> S"
+    and "i \<in> T \<rightarrow> S"
     and contr: "continuous_on S r"
-    and "r ` S \<subseteq> T"
+    and "r \<in> S \<rightarrow> T"
     and ri: "\<And>y. y \<in> T \<Longrightarrow> r (i y) = y"
-    and FP: "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>x\<in>S. f x = x"
+    and FP: "\<And>f. \<lbrakk>continuous_on S f; f \<in> S \<rightarrow> S\<rbrakk> \<Longrightarrow> \<exists>x\<in>S. f x = x"
     and contg: "continuous_on T g"
-    and "g ` T \<subseteq> T"
+    and "g \<in> T \<rightarrow> T"
   obtains y where "y \<in> T" and "g y = y"
 proof -
   have "\<exists>x\<in>S. (i \<circ> g \<circ> r) x = x"
   proof (rule FP)
     show "continuous_on S (i \<circ> g \<circ> r)"
-      by (meson contt contr assms(4) contg assms(8) continuous_on_compose continuous_on_subset)
-    show "(i \<circ> g \<circ> r) ` S \<subseteq> S"
+      by (metis assms(4) assms(8) contg continuous_on_compose continuous_on_subset contr contt funcset_image)
+    show "(i \<circ> g \<circ> r) \<in> S \<rightarrow> S"
       using assms(2,4,8) by force
   qed
   then obtain x where x: "x \<in> S" "(i \<circ> g \<circ> r) x = x" ..
@@ -928,8 +928,8 @@
   fixes S :: "'a::topological_space set"
     and T :: "'b::topological_space set"
   assumes "S homeomorphic T"
-  shows "(\<forall>f. continuous_on S f \<and> f ` S \<subseteq> S \<longrightarrow> (\<exists>x\<in>S. f x = x)) \<longleftrightarrow>
-         (\<forall>g. continuous_on T g \<and> g ` T \<subseteq> T \<longrightarrow> (\<exists>y\<in>T. g y = y))"
+  shows "(\<forall>f. continuous_on S f \<and> f \<in> S \<rightarrow> S \<longrightarrow> (\<exists>x\<in>S. f x = x)) \<longleftrightarrow>
+         (\<forall>g. continuous_on T g \<and> g \<in> T \<rightarrow> T \<longrightarrow> (\<exists>y\<in>T. g y = y))"
          (is "?lhs = ?rhs")
 proof -
   obtain r i where r:
@@ -940,11 +940,11 @@
   proof
     assume ?lhs
     with r show ?rhs
-      by (metis invertible_fixpoint_property[of T i S r] order_refl)
+      by (smt (verit, del_insts) Pi_iff image_eqI invertible_fixpoint_property[of T i S r])
   next
     assume ?rhs
     with r show ?lhs
-      by (metis invertible_fixpoint_property[of S r T i] order_refl)
+      by (smt (verit, del_insts) Pi_iff image_eqI invertible_fixpoint_property[of S r T i])
   qed
 qed
 
@@ -952,9 +952,9 @@
   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
     and S :: "'a set"
   assumes "T retract_of S"
-    and FP: "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>x\<in>S. f x = x"
+    and FP: "\<And>f. \<lbrakk>continuous_on S f; f \<in> S \<rightarrow> S\<rbrakk> \<Longrightarrow> \<exists>x\<in>S. f x = x"
     and contg: "continuous_on T g"
-    and "g ` T \<subseteq> T"
+    and "g \<in> T \<rightarrow> T"
   obtains y where "y \<in> T" and "g y = y"
 proof -
   obtain h where "retraction S T h"
@@ -962,23 +962,24 @@
   then show ?thesis
     unfolding retraction_def
     using invertible_fixpoint_property[OF continuous_on_id _ _ _ _ FP]
-    by (metis assms(4) contg image_ident that)
+    by (smt (verit, del_insts) Pi_iff assms(4) contg subsetD that)
 qed
 
 lemma retraction:
-  "retraction S T r \<longleftrightarrow>
-    T \<subseteq> S \<and> continuous_on S r \<and> r ` S = T \<and> (\<forall>x \<in> T. r x = x)"
-  by (force simp: retraction_def)
+  "retraction S T r \<longleftrightarrow> T \<subseteq> S \<and> continuous_on S r \<and> r ` S = T \<and> (\<forall>x \<in> T. r x = x)"
+  by (force simp: retraction_def simp flip: image_subset_iff_funcset)
 
 lemma retractionE: \<comment> \<open>yields properties normalized wrt. simp -- less likely to loop\<close>
   assumes "retraction S T r"
-  obtains "T = r ` S" "r ` S \<subseteq> S" "continuous_on S r" "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x"
+  obtains "T = r ` S" "r \<in> S \<rightarrow> S" "continuous_on S r" "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x"
 proof (rule that)
   from retraction [of S T r] assms
   have "T \<subseteq> S" "continuous_on S r" "r ` S = T" and "\<forall>x \<in> T. r x = x"
     by simp_all
-  then show "T = r ` S" "r ` S \<subseteq> S" "continuous_on S r"
-    by simp_all
+  then show  "r \<in> S \<rightarrow> S" "continuous_on S r"
+    by auto
+  then show "T = r ` S"
+    using \<open>r ` S = T\<close> by blast
   from \<open>\<forall>x \<in> T. r x = x\<close> have "r x = x" if "x \<in> T" for x
     using that by simp
   with \<open>r ` S = T\<close> show "r (r x) = r x" if "x \<in> S" for x
@@ -987,7 +988,7 @@
 
 lemma retract_ofE: \<comment> \<open>yields properties normalized wrt. simp -- less likely to loop\<close>
   assumes "T retract_of S"
-  obtains r where "T = r ` S" "r ` S \<subseteq> S" "continuous_on S r" "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x"
+  obtains r where "T = r ` S" "r \<in> S \<rightarrow> S" "continuous_on S r" "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x"
 proof -
   from assms obtain r where "retraction S T r"
     by (auto simp add: retract_of_def)
@@ -996,30 +997,31 @@
 qed
 
 lemma retract_of_imp_extensible:
-  assumes "S retract_of T" and "continuous_on S f" and "f ` S \<subseteq> U"
-  obtains g where "continuous_on T g" "g ` T \<subseteq> U" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+  assumes "S retract_of T" and "continuous_on S f" and "f \<in> S \<rightarrow> U"
+  obtains g where "continuous_on T g" "g \<in> T \<rightarrow> U" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
 proof -
   from \<open>S retract_of T\<close> obtain r where "retraction T S r"
     by (auto simp add: retract_of_def)
-  show thesis
-    by (rule that [of "f \<circ> r"])
-      (use \<open>continuous_on S f\<close> \<open>f ` S \<subseteq> U\<close> \<open>retraction T S r\<close> in \<open>auto simp: continuous_on_compose2 retraction\<close>)
+  then have "continuous_on T (f \<circ> r)"
+      by (metis assms(2) continuous_on_compose retraction)
+  then show thesis
+    by (smt (verit, best) Pi_iff \<open>retraction T S r\<close> assms(3) comp_apply retraction_def that)
 qed
 
 lemma idempotent_imp_retraction:
-  assumes "continuous_on S f" and "f ` S \<subseteq> S" and "\<And>x. x \<in> S \<Longrightarrow> f(f x) = f x"
+  assumes "continuous_on S f" and "f \<in> S \<rightarrow> S" and "\<And>x. x \<in> S \<Longrightarrow> f(f x) = f x"
     shows "retraction S (f ` S) f"
-by (simp add: assms retraction)
+  by (simp add: assms funcset_image retraction)
 
 lemma retraction_subset:
-  assumes "retraction S T r" and "T \<subseteq> s'" and "s' \<subseteq> S"
-  shows "retraction s' T r"
+  assumes "retraction S T r" and "T \<subseteq> S'" and "S' \<subseteq> S"
+  shows "retraction S' T r"
   unfolding retraction_def
-  by (metis assms continuous_on_subset image_mono retraction)
+  by (metis assms continuous_on_subset image_mono image_subset_iff_funcset retraction)
 
 lemma retract_of_subset:
-  assumes "T retract_of S" and "T \<subseteq> s'" and "s' \<subseteq> S"
-    shows "T retract_of s'"
+  assumes "T retract_of S" and "T \<subseteq> S'" and "S' \<subseteq> S"
+    shows "T retract_of S'"
 by (meson assms retract_of_def retraction_subset)
 
 lemma retraction_refl [simp]: "retraction S S (\<lambda>x. x)"
@@ -1054,10 +1056,10 @@
   assumes "S retract_of T"
     shows "closedin (top_of_set T) S"
 proof -
-  obtain r where r: "S \<subseteq> T" "continuous_on T r" "r ` T \<subseteq> S" "\<And>x. x \<in> S \<Longrightarrow> r x = x"
+  obtain r where r: "S \<subseteq> T" "continuous_on T r" "r \<in> T \<rightarrow> S" "\<And>x. x \<in> S \<Longrightarrow> r x = x"
     using assms by (auto simp: retract_of_def retraction_def)
   have "S = {x\<in>T. x = r x}"
-    using r by auto
+    using r by force
   also have "\<dots> = T \<inter> ((\<lambda>x. (x, r x)) -` ({y. \<exists>x. y = (x, x)}))"
     unfolding vimage_def mem_Times_iff fst_conv snd_conv
     using r
@@ -1088,14 +1090,14 @@
   shows "openin (top_of_set S) (S \<inter> r -` U) \<longleftrightarrow> openin (top_of_set T) U" (is "?lhs = ?rhs")
 proof
   show "?lhs \<Longrightarrow> ?rhs"
-    using r retraction_def retractionE
-    by (smt (verit, best) continuous_right_inverse_imp_quotient_map retraction_subset \<open>U \<subseteq> T\<close>)
+    using r
+    by (smt (verit, del_insts) assms(2) continuous_right_inverse_imp_quotient_map image_subset_iff_funcset r retractionE retraction_def retraction_subset)
   show "?rhs \<Longrightarrow> ?lhs"
-    by (meson continuous_openin_preimage r retraction_def)
+    by (metis continuous_on_open r retraction)
 qed
 
 lemma retract_of_Times:
-   "\<lbrakk>S retract_of s'; T retract_of t'\<rbrakk> \<Longrightarrow> (S \<times> T) retract_of (s' \<times> t')"
+   "\<lbrakk>S retract_of S'; T retract_of T'\<rbrakk> \<Longrightarrow> (S \<times> T) retract_of (S' \<times> T')"
 apply (simp add: retract_of_def retraction_def Sigma_mono, clarify)
 apply (rename_tac f g)
 apply (rule_tac x="\<lambda>z. ((f \<circ> fst) z, (g \<circ> snd) z)" in exI)
@@ -1238,7 +1240,7 @@
 lemma path_finish_in_topspace: "pathin X g \<Longrightarrow> g 1 \<in> topspace X"
   by (force simp: pathin_def continuous_map)
 
-lemma path_image_subset_topspace: "pathin X g \<Longrightarrow> g ` ({0..1}) \<subseteq> topspace X"
+lemma path_image_subset_topspace: "pathin X g \<Longrightarrow> g \<in> ({0..1}) \<rightarrow> topspace X"
   by (force simp: pathin_def continuous_map)
 
 definition path_connected_space :: "'a topology \<Rightarrow> bool"
@@ -1262,9 +1264,9 @@
 lemma path_connectedin:
      "path_connectedin X S \<longleftrightarrow>
         S \<subseteq> topspace X \<and>
-        (\<forall>x \<in> S. \<forall>y \<in> S. \<exists>g. pathin X g \<and> g ` {0..1} \<subseteq> S \<and> g 0 = x \<and> g 1 = y)"
+        (\<forall>x \<in> S. \<forall>y \<in> S. \<exists>g. pathin X g \<and> g \<in> {0..1} \<rightarrow> S \<and> g 0 = x \<and> g 1 = y)"
   unfolding path_connectedin_def path_connected_space_def pathin_def continuous_map_in_subtopology
-  by (intro conj_cong refl ball_cong) (simp_all add: inf.absorb_iff2)
+  by (intro conj_cong refl ball_cong) (simp_all add: inf.absorb_iff2 flip: image_subset_iff_funcset)
 
 lemma path_connectedin_topspace:
      "path_connectedin X (topspace X) \<longleftrightarrow> path_connected_space X"
@@ -1310,21 +1312,21 @@
   assumes f: "continuous_map X Y f" and S: "path_connectedin X S"
   shows "path_connectedin Y (f ` S)"
 proof -
-  have fX: "f ` (topspace X) \<subseteq> topspace Y"
-    by (metis f continuous_map_image_subset_topspace)
+  have fX: "f \<in> (topspace X) \<rightarrow> topspace Y"
+    using continuous_map_def f by fastforce
   show ?thesis
     unfolding path_connectedin
   proof (intro conjI ballI; clarify?)
     fix x
     assume "x \<in> S"
     show "f x \<in> topspace Y"
-      by (meson S fX \<open>x \<in> S\<close> image_subset_iff path_connectedin_subset_topspace set_mp)
+      using S \<open>x \<in> S\<close> fX path_connectedin_subset_topspace by fastforce
   next
     fix x y
     assume "x \<in> S" and "y \<in> S"
-    then obtain g where g: "pathin X g" "g ` {0..1} \<subseteq> S" "g 0 = x" "g 1 = y"
+    then obtain g where g: "pathin X g" "g \<in> {0..1} \<rightarrow> S" "g 0 = x" "g 1 = y"
       using S  by (force simp: path_connectedin)
-    show "\<exists>g. pathin Y g \<and> g ` {0..1} \<subseteq> f ` S \<and> g 0 = f x \<and> g 1 = f y"
+    show "\<exists>g. pathin Y g \<and> g \<in> {0..1} \<rightarrow> f ` S \<and> g 0 = f x \<and> g 1 = f y"
     proof (intro exI conjI)
       show "pathin Y (f \<circ> g)"
         using \<open>pathin X g\<close> f pathin_compose by auto
@@ -1361,7 +1363,7 @@
   shows "path_connectedin Y (f ` U) \<longleftrightarrow> path_connectedin X U"
   unfolding path_connectedin_def
 proof (intro conj_cong homeomorphic_path_connected_space)
-  show "(f ` U \<subseteq> topspace Y) = (U \<subseteq> topspace X)"
+  show "f ` U \<subseteq> topspace Y \<longleftrightarrow> (U \<subseteq> topspace X)"
     using assms homeomorphic_imp_surjective_map by blast
 next
   assume "U \<subseteq> topspace X"
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Brouwer_Fixpoint.thy
--- a/src/HOL/Analysis/Brouwer_Fixpoint.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Brouwer_Fixpoint.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -24,7 +24,7 @@
   assumes "contractible T" "S retract_of T"
     shows "contractible S"
 using assms
-apply (clarsimp simp add: retract_of_def contractible_def retraction_def homotopic_with)
+apply (clarsimp simp add: retract_of_def contractible_def retraction_def homotopic_with image_subset_iff_funcset)
 apply (rule_tac x="r a" in exI)
 apply (rule_tac x="r \<circ> h" in exI)
 apply (intro conjI continuous_intros continuous_on_compose)
@@ -126,8 +126,8 @@
     by (metis eq_id_iff homotopic_with_id2 topspace_euclidean_subtopology)
   ultimately
   show ?thesis
-    unfolding homotopy_equivalent_space_def 
-    by (metis (no_types, lifting) continuous_map_subtopology_eu continuous_on_id' id_def image_id r retraction_def)
+    unfolding homotopy_equivalent_space_def
+    by (smt (verit, del_insts) continuous_map_id continuous_map_subtopology_eu id_def r retraction retraction_comp subset_refl) 
 qed
 
 lemma deformation_retract:
@@ -1807,7 +1807,7 @@
     and "convex S"
     and "interior S \<noteq> {}"
     and "continuous_on S f"
-    and "f ` S \<subseteq> S"
+    and "f \<in> S \<rightarrow> S"
   obtains x where "x \<in> S" and "f x = x"
 proof -
   let ?U = "cbox 0 One :: 'a set"
@@ -1824,8 +1824,7 @@
   then have *: "interior ?U \<noteq> {}" by fast
   have *: "?U homeomorphic S"
     using homeomorphic_convex_compact[OF convex_box(1) compact_cbox * assms(2,1,3)] .
-  have "\<forall>f. continuous_on ?U f \<and> f ` ?U \<subseteq> ?U \<longrightarrow>
-    (\<exists>x\<in>?U. f x = x)"
+  have "\<forall>f. continuous_on ?U f \<and> f \<in> ?U \<rightarrow> ?U \<longrightarrow> (\<exists>x\<in>?U. f x = x)"
     using brouwer_cube by auto
   then show ?thesis
     unfolding homeomorphic_fixpoint_property[OF *]
@@ -1839,7 +1838,7 @@
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes "e > 0"
     and "continuous_on (cball a e) f"
-    and "f ` cball a e \<subseteq> cball a e"
+    and "f \<in> cball a e \<rightarrow> cball a e"
   obtains x where "x \<in> cball a e" and "f x = x"
   using brouwer_weak[OF compact_cball convex_cball, of a e f]
   unfolding interior_cball ball_eq_empty
@@ -1851,7 +1850,7 @@
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes S: "compact S" "convex S" "S \<noteq> {}"
     and contf: "continuous_on S f"
-    and fim: "f ` S \<subseteq> S"
+    and fim: "f \<in> S \<rightarrow> S"
   obtains x where "x \<in> S" and "f x = x"
 proof -
   have "\<exists>e>0. S \<subseteq> cball 0 e"
@@ -1864,12 +1863,12 @@
     show "continuous_on (cball 0 e) (f \<circ> closest_point S)"
       by (meson assms closest_point_in_set compact_eq_bounded_closed contf continuous_on_closest_point
           continuous_on_compose continuous_on_subset image_subsetI)
-    show "(f \<circ> closest_point S) ` cball 0 e \<subseteq> cball 0 e"
-      by clarsimp (metis S fim closest_point_exists(1) compact_eq_bounded_closed e(2) image_subset_iff mem_cball_0 subsetCE)
+    show "f \<circ> closest_point S \<in> cball 0 e \<rightarrow> cball 0 e"
+      by (smt (verit) Pi_iff assms(1) assms(3) closest_point_in_set comp_apply compact_eq_bounded_closed e(2) fim subset_eq)
   qed (use assms in auto)
   then obtain x where x: "x \<in> cball 0 e" "(f \<circ> closest_point S) x = x" ..
-  have "x \<in> S"
-    by (metis closest_point_in_set comp_apply compact_imp_closed fim image_eqI S(1) S(3) subset_iff x(2))
+  with S have "x \<in> S"
+    by (metis PiE closest_point_in_set comp_apply compact_imp_closed fim)
   then have *: "closest_point S x = x"
     by (rule closest_point_self)
   show thesis
@@ -1877,7 +1876,7 @@
     show "closest_point S x \<in> S"
       by (simp add: "*" \<open>x \<in> S\<close>)
     show "f (closest_point S x) = closest_point S x"
-      using "*" x(2) by auto
+      using "*" x by auto
   qed
 qed
 
@@ -1897,14 +1896,13 @@
   proof (rule retract_fixpoint_property[OF *, of "\<lambda>x. scaleR 2 a - x"])
     show "continuous_on (frontier (cball a e)) ((-) (2 *\<^sub>R a))"
       by (intro continuous_intros)
-    show "(-) (2 *\<^sub>R a) ` frontier (cball a e) \<subseteq> frontier (cball a e)"
+    show "(-) (2 *\<^sub>R a) \<in> frontier (cball a e) \<rightarrow> frontier (cball a e)"
       by clarsimp (metis "**" dist_norm norm_minus_cancel)
   qed (auto simp: dist_norm intro: brouwer_ball[OF assms])
   then have "scaleR 2 a = scaleR 1 x + scaleR 1 x"
     by (auto simp: algebra_simps)
   then have "a = x"
-    unfolding scaleR_left_distrib[symmetric]
-    by auto
+    unfolding scaleR_left_distrib[symmetric] by auto
   then show False
     using x assms by auto
 qed
@@ -1977,7 +1975,7 @@
   fixes S :: "'a::euclidean_space set"
   assumes "bounded S" and fros: "frontier S \<subseteq> T"
       and contf: "continuous_on (closure S) f"
-      and fim: "f ` S \<subseteq> T"
+      and fim: "f \<in> S \<rightarrow> T"
       and fid: "\<And>x. x \<in> T \<Longrightarrow> f x = x"
     shows "S \<subseteq> T"
 proof (rule ccontr)
@@ -1995,7 +1993,7 @@
     using \<open>bounded S\<close> bounded_subset_ballD by blast
   have notga: "g x \<noteq> a" for x
     unfolding g_def using fros fim \<open>a \<notin> T\<close>
-    by (metis Un_iff \<open>a \<in> S\<close> closure_Un_frontier fid imageI subset_eq)
+    by (metis PiE Un_iff \<open>a \<in> S\<close> closure_Un_frontier fid subsetD)
   define h where "h \<equiv> (\<lambda>y. a + (B / norm(y - a)) *\<^sub>R (y - a)) \<circ> g"
   have "\<not> (frontier (cball a B) retract_of (cball a B))"
     by (metis no_retraction_cball \<open>0 < B\<close>)
@@ -2009,12 +2007,12 @@
     show "continuous_on (cball a B) h"
       unfolding h_def
       by (intro continuous_intros) (use contg continuous_on_subset notga in auto)
-    show "h ` cball a B \<subseteq> frontier (cball a B)"
+    show "h \<in> cball a B \<rightarrow> frontier (cball a B)"
       using \<open>0 < B\<close> by (auto simp: h_def notga dist_norm)
     show "\<And>x. x \<in> frontier (cball a B) \<Longrightarrow> h x = x"
-      apply (auto simp: h_def algebra_simps)
-      apply (simp add: vector_add_divide_simps  notga)
-      by (metis (no_types, opaque_lifting) B add.commute dist_commute  dist_norm g_def mem_ball not_less_iff_gr_or_eq  subset_eq)
+      using notga \<open>0 < B\<close>
+      apply (simp add: g_def h_def field_simps)
+      by (metis B dist_commute dist_norm mem_ball order_less_irrefl subset_eq)
   qed
   ultimately show False by simp
 qed
@@ -2131,7 +2129,7 @@
         using arelS relS rel_frontier_def by fastforce
       show "continuous_on (T - {a}) (\<lambda>x. a + dd (x - a) *\<^sub>R (x - a))"
         using contdd by (simp add: o_def)
-      show "(\<lambda>x. a + dd (x - a) *\<^sub>R (x - a)) ` (T - {a}) \<subseteq> rel_frontier S"
+      show "(\<lambda>x. a + dd (x - a) *\<^sub>R (x - a)) \<in> (T - {a}) \<rightarrow> rel_frontier S"
         apply (auto simp: rel_frontier_def)
         apply (metis Diff_subset add.commute affS dd1 diff_add_cancel eq_iff_diff_eq_0 rel_frontier_def subset_iff)
         by (metis DiffE add.commute affS dd1 diff_add_cancel eq_iff_diff_eq_0 rel_frontier_def rev_subsetD)
@@ -2216,20 +2214,25 @@
 subsubsection\<open>Borsuk-style characterization of separation\<close>
 
 lemma continuous_on_Borsuk_map:
-   "a \<notin> s \<Longrightarrow>  continuous_on s (\<lambda>x. inverse(norm (x - a)) *\<^sub>R (x - a))"
+   "a \<notin> S \<Longrightarrow>  continuous_on S (\<lambda>x. inverse(norm (x - a)) *\<^sub>R (x - a))"
 by (rule continuous_intros | force)+
 
 lemma Borsuk_map_into_sphere:
-   "(\<lambda>x. inverse(norm (x - a)) *\<^sub>R (x - a)) ` s \<subseteq> sphere 0 1 \<longleftrightarrow> (a \<notin> s)"
-  by auto (metis eq_iff_diff_eq_0 left_inverse norm_eq_zero)
+   "(\<lambda>x. inverse(norm (x - a)) *\<^sub>R (x - a)) \<in> S \<rightarrow> sphere 0 1 \<longleftrightarrow> (a \<notin> S)"
+proof -
+  have "\<And>x. \<lbrakk>a \<notin> S; x \<in> S\<rbrakk> \<Longrightarrow> inverse (norm (x - a)) * norm (x - a) = 1"
+    by (metis left_inverse norm_eq_zero right_minus_eq)
+  then show ?thesis
+    by force
+qed
 
 lemma Borsuk_maps_homotopic_in_path_component:
-  assumes "path_component (- s) a b"
-    shows "homotopic_with_canon (\<lambda>x. True) s (sphere 0 1)
+  assumes "path_component (- S) a b"
+    shows "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1)
                    (\<lambda>x. inverse(norm(x - a)) *\<^sub>R (x - a))
                    (\<lambda>x. inverse(norm(x - b)) *\<^sub>R (x - b))"
 proof -
-  obtain g where "path g" "path_image g \<subseteq> -s" "pathstart g = a" "pathfinish g = b"
+  obtain g where "path g" "path_image g \<subseteq> -S" "pathstart g = a" "pathfinish g = b"
     using assms by (auto simp: path_component_def)
   then show ?thesis
     apply (simp add: path_def path_image_def pathstart_def pathfinish_def homotopic_with_def)
@@ -2240,61 +2243,59 @@
 
 lemma non_extensible_Borsuk_map:
   fixes a :: "'a :: euclidean_space"
-  assumes "compact s" and cin: "c \<in> components(- s)" and boc: "bounded c" and "a \<in> c"
-    shows "\<not> (\<exists>g. continuous_on (s \<union> c) g \<and>
-                  g ` (s \<union> c) \<subseteq> sphere 0 1 \<and>
-                  (\<forall>x \<in> s. g x = inverse(norm(x - a)) *\<^sub>R (x - a)))"
+  assumes "compact S" and cin: "C \<in> components(- S)" and boc: "bounded C" and "a \<in> C"
+    shows "\<not> (\<exists>g. continuous_on (S \<union> C) g \<and>
+                  g \<in> (S \<union> C) \<rightarrow> sphere 0 1 \<and>
+                  (\<forall>x \<in> S. g x = inverse(norm(x - a)) *\<^sub>R (x - a)))"
 proof -
-  have "closed s" using assms by (simp add: compact_imp_closed)
-  have "c \<subseteq> -s"
+  have "closed S" using assms by (simp add: compact_imp_closed)
+  have "C \<subseteq> -S"
     using assms by (simp add: in_components_subset)
-  with \<open>a \<in> c\<close> have "a \<notin> s" by blast
-  then have ceq: "c = connected_component_set (- s) a"
-    by (metis \<open>a \<in> c\<close> cin components_iff connected_component_eq)
-  then have "bounded (s \<union> connected_component_set (- s) a)"
-    using \<open>compact s\<close> boc compact_imp_bounded by auto
-  with bounded_subset_ballD obtain r where "0 < r" and r: "(s \<union> connected_component_set (- s) a) \<subseteq> ball a r"
+  with \<open>a \<in> C\<close> have "a \<notin> S" by blast
+  then have ceq: "C = connected_component_set (- S) a"
+    by (metis \<open>a \<in> C\<close> cin components_iff connected_component_eq)
+  then have "bounded (S \<union> connected_component_set (- S) a)"
+    using \<open>compact S\<close> boc compact_imp_bounded by auto
+  with bounded_subset_ballD obtain r where "0 < r" and r: "(S \<union> connected_component_set (- S) a) \<subseteq> ball a r"
     by blast
   { fix g
-    assume "continuous_on (s \<union> c) g"
-            "g ` (s \<union> c) \<subseteq> sphere 0 1"
-       and [simp]: "\<And>x. x \<in> s \<Longrightarrow> g x = (x - a) /\<^sub>R norm (x - a)"
-    then have [simp]: "\<And>x. x \<in> s \<union> c \<Longrightarrow> norm (g x) = 1"
+    assume "continuous_on (S \<union> C) g"
+            "g \<in> (S \<union> C) \<rightarrow> sphere 0 1"
+       and [simp]: "\<And>x. x \<in> S \<Longrightarrow> g x = (x - a) /\<^sub>R norm (x - a)"
+    then have norm_g1[simp]: "\<And>x. x \<in> S \<union> C \<Longrightarrow> norm (g x) = 1"
       by force
-    have cb_eq: "cball a r = (s \<union> connected_component_set (- s) a) \<union>
-                      (cball a r - connected_component_set (- s) a)"
+    have cb_eq: "cball a r = (S \<union> connected_component_set (- S) a) \<union>
+                      (cball a r - connected_component_set (- S) a)"
       using ball_subset_cball [of a r] r by auto
-    have cont1: "continuous_on (s \<union> connected_component_set (- s) a)
+    have cont1: "continuous_on (S \<union> connected_component_set (- S) a)
                      (\<lambda>x. a + r *\<^sub>R g x)"
-      using \<open>continuous_on (s \<union> c) g\<close> ceq
+      using \<open>continuous_on (S \<union> C) g\<close> ceq
       by (intro continuous_intros) blast
-    have cont2: "continuous_on (cball a r - connected_component_set (- s) a)
+    have cont2: "continuous_on (cball a r - connected_component_set (- S) a)
             (\<lambda>x. a + r *\<^sub>R ((x - a) /\<^sub>R norm (x - a)))"
-      by (rule continuous_intros | force simp: \<open>a \<notin> s\<close>)+
+      by (rule continuous_intros | force simp: \<open>a \<notin> S\<close>)+
     have 1: "continuous_on (cball a r)
-             (\<lambda>x. if connected_component (- s) a x
+             (\<lambda>x. if connected_component (- S) a x
                   then a + r *\<^sub>R g x
                   else a + r *\<^sub>R ((x - a) /\<^sub>R norm (x - a)))"
       apply (subst cb_eq)
       apply (rule continuous_on_cases [OF _ _ cont1 cont2])
-      using \<open>closed s\<close> ceq cin 
+      using \<open>closed S\<close> ceq cin 
       by (force simp: closed_Diff open_Compl closed_Un_complement_component open_connected_component)+
-    have 2: "(\<lambda>x. a + r *\<^sub>R g x) ` (cball a r \<inter> connected_component_set (- s) a)
+    have 2: "(\<lambda>x. a + r *\<^sub>R g x) ` (cball a r \<inter> connected_component_set (- S) a)
              \<subseteq> sphere a r "
       using \<open>0 < r\<close> by (force simp: dist_norm ceq)
     have "retraction (cball a r) (sphere a r)
-            (\<lambda>x. if x \<in> connected_component_set (- s) a
+            (\<lambda>x. if x \<in> connected_component_set (- S) a
                  then a + r *\<^sub>R g x
                  else a + r *\<^sub>R ((x - a) /\<^sub>R norm (x - a)))"
-      using  \<open>0 < r\<close>
-      apply (simp add: retraction_def dist_norm 1 2, safe)
-      apply (force simp: dist_norm abs_if mult_less_0_iff divide_simps \<open>a \<notin> s\<close>)
-      using r
-      by (auto simp: dist_norm norm_minus_commute)
+      using  \<open>0 < r\<close> \<open>a \<notin> S\<close> \<open>a \<in> C\<close> r
+      by (auto simp: norm_minus_commute retraction_def Pi_iff ceq dist_norm abs_if 
+          mult_less_0_iff divide_simps 1 2)
     then have False
       using no_retraction_cball
              [OF \<open>0 < r\<close>, of a, unfolded retract_of_def, simplified, rule_format,
-              of "\<lambda>x. if x \<in> connected_component_set (- s) a
+              of "\<lambda>x. if x \<in> connected_component_set (- S) a
                       then a + r *\<^sub>R g x
                       else a + r *\<^sub>R inverse(norm(x - a)) *\<^sub>R (x - a)"]
       by blast
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Fashoda_Theorem.thy
--- a/src/HOL/Analysis/Fashoda_Theorem.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Fashoda_Theorem.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -174,8 +174,8 @@
       "(negatex \<circ> sqprojection \<circ> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w)) x = x"
     apply (rule brouwer_weak[of "cbox (- 1) (1::real^2)" "negatex \<circ> sqprojection \<circ> ?F"])
     apply (rule compact_cbox convex_box)+
-    unfolding interior_cbox
-    apply (rule 1 2 3)+
+    unfolding interior_cbox image_subset_iff_funcset [symmetric]
+       apply (rule 1 2 3)+
     apply blast
     done
   have "?F x \<noteq> 0"
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Further_Topology.thy
--- a/src/HOL/Analysis/Further_Topology.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Further_Topology.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -231,7 +231,7 @@
   qed
   have conT0: "continuous_on (T - {0}) (\<lambda>y. inverse(norm y) *\<^sub>R y)"
     by (intro continuous_intros) auto
-  have sub0T: "(\<lambda>y. y /\<^sub>R norm y) ` (T - {0}) \<subseteq> sphere 0 1 \<inter> T"
+  have sub0T: "(\<lambda>y. y /\<^sub>R norm y) \<in> (T - {0}) \<rightarrow> sphere 0 1 \<inter> T"
     by (fastforce simp: assms(2) subspace_mul)
   obtain c where homhc: "homotopic_with_canon (\<lambda>z. True) (sphere 0 1 \<inter> S) (sphere 0 1 \<inter> T) h (\<lambda>x. c)"
   proof
@@ -307,7 +307,7 @@
   assumes "convex S" "bounded S" "convex T" "bounded T"
       and affST: "aff_dim S < aff_dim T"
       and contf: "continuous_on (rel_frontier S) f"
-      and fim: "f ` (rel_frontier S) \<subseteq> rel_frontier T"
+      and fim: "f \<in> (rel_frontier S) \<rightarrow> rel_frontier T"
   obtains c where "homotopic_with_canon (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
 proof (cases "S = {}")
   case True
@@ -319,7 +319,7 @@
   proof (cases "T = {}")
     case True
     then show ?thesis
-      using fim that by auto
+      using fim that by (simp add: Pi_iff)
   next
     case False
     obtain T':: "'a set"
@@ -345,7 +345,7 @@
     have "\<exists>c. homotopic_with_canon (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
       using spheremap_lemma2 homotopy_eqv_cohomotopic_triviality_null[OF relS] 
       using homotopy_eqv_homotopic_triviality_null_imp [OF relT contf fim]
-      by (metis \<open>S' \<subseteq> T'\<close> \<open>subspace S'\<close> \<open>subspace T'\<close> dimST')
+      by (metis \<open>S' \<subseteq> T'\<close> \<open>subspace S'\<close> \<open>subspace T'\<close> dimST' image_subset_iff_funcset)
     with that show ?thesis by blast
   qed
 qed
@@ -353,7 +353,7 @@
 lemma inessential_spheremap_lowdim:
   fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
   assumes
-   "DIM('M) < DIM('a)" and f: "continuous_on (sphere a r) f" "f ` (sphere a r) \<subseteq> (sphere b s)"
+   "DIM('M) < DIM('a)" and f: "continuous_on (sphere a r) f" "f \<in> (sphere a r) \<rightarrow> (sphere b s)"
    obtains c where "homotopic_with_canon (\<lambda>z. True) (sphere a r) (sphere b s) f (\<lambda>x. c)"
 proof (cases "s \<le> 0")
   case True then show ?thesis
@@ -505,7 +505,7 @@
           using \<open>C \<in> \<F>\<close> \<open>D face_of C\<close> face_of_polytope_polytope poly polytope_imp_polyhedron by auto
         ultimately have relf_sub: "rel_frontier D \<subseteq> \<Union> {E. E face_of C \<and> aff_dim E < p}"
           by (simp add: rel_frontier_of_polyhedron Union_mono)
-        then have him_relf: "h ` rel_frontier D \<subseteq> rel_frontier T"
+        then have him_relf: "h \<in> rel_frontier D \<rightarrow> rel_frontier T"
           using \<open>C \<in> \<F>\<close> him by blast
         have "convex D"
           by (simp add: \<open>polyhedron D\<close> polyhedron_imp_convex)
@@ -516,9 +516,9 @@
         then have *: "(\<exists>c. homotopic_with_canon (\<lambda>x. True) (rel_frontier D) (rel_frontier T) h (\<lambda>x. c)) =
                       (\<exists>g. continuous_on UNIV g \<and>  range g \<subseteq> rel_frontier T \<and>
                            (\<forall>x\<in>rel_frontier D. g x = h x))"
-          by (simp add: assms rel_frontier_eq_empty him_relf nullhomotopic_into_rel_frontier_extension [OF closed_rel_frontier])
-        have "(\<exists>c. homotopic_with_canon (\<lambda>x. True) (rel_frontier D) (rel_frontier T) h (\<lambda>x. c))"
-          by (metis inessential_spheremap_lowdim_gen
+          by (simp add: assms image_subset_iff_funcset rel_frontier_eq_empty him_relf nullhomotopic_into_rel_frontier_extension [OF closed_rel_frontier])
+        have "\<exists>c. homotopic_with_canon (\<lambda>x. True) (rel_frontier D) (rel_frontier T) h (\<lambda>x. c)"
+          by (metis inessential_spheremap_lowdim_gen 
                  [OF \<open>convex D\<close> \<open>bounded D\<close> \<open>convex T\<close> \<open>bounded T\<close> affD_lessT contDh him_relf])
         then obtain g where contg: "continuous_on UNIV g"
                         and gim: "range g \<subseteq> rel_frontier T"
@@ -852,11 +852,11 @@
       and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
       and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X < aff_dim T"
       and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X"
-      and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
+      and contf: "continuous_on S f" and fim: "f \<in> S \<rightarrow> rel_frontier T"
   obtains g where "continuous_on (\<Union>\<F>) g"
      "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
 proof -
-  obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+  obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g \<in> V \<rightarrow> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
     using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
   have "compact S"
     by (meson assms compact_Union poly polytope_imp_compact seq_compact_closed_subset seq_compact_eq_compact)
@@ -906,11 +906,11 @@
       and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
       and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T"
       and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X"
-      and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
+      and contf: "continuous_on S f" and fim: "f \<in> S \<rightarrow> rel_frontier T"
   obtains C g where "finite C" "disjnt C S" "continuous_on (\<Union>\<F> - C) g"
      "g ` (\<Union>\<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
 proof -
-  obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+  obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g \<in> V \<rightarrow> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
     using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
   have "compact S"
     by (meson assms compact_Union poly polytope_imp_compact seq_compact_closed_subset seq_compact_eq_compact)
@@ -974,13 +974,13 @@
   assumes "compact S" "convex U" "bounded U"
       and aff: "aff_dim T \<le> aff_dim U"
       and "S \<subseteq> T" and contf: "continuous_on S f"
-      and fim: "f ` S \<subseteq> rel_frontier U"
+      and fim: "f \<in> S \<rightarrow> rel_frontier U"
  obtains K g where "finite K" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
-                   "g ` (T - K) \<subseteq> rel_frontier U"
+                   "g \<in> (T - K) \<rightarrow> rel_frontier U"
                    "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
 proof -
   have "\<exists>K g. finite K \<and> disjnt K S \<and> continuous_on (T - K) g \<and>
-              g ` (T - K) \<subseteq> rel_frontier U \<and> (\<forall>x \<in> S. g x = f x)"
+              g \<in> (T - K) \<rightarrow> rel_frontier U \<and> (\<forall>x \<in> S. g x = f x)"
        if "affine T" "S \<subseteq> T" and aff: "aff_dim T \<le> aff_dim U"  for T
   proof (cases "S = {}")
     case True
@@ -1089,7 +1089,8 @@
         by (metis image_comp image_mono cpt_subset)
       also have "... \<subseteq> rel_frontier U"
         by (rule gim)
-      finally show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K) \<subseteq> rel_frontier U" .
+      finally show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) \<in> (T - K) \<rightarrow> rel_frontier U"
+        by blast
       show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) x = f x" if "x \<in> S" for x
       proof -
         have "(g \<circ> closest_point (cbox (- c) c \<inter> T)) x = g x"
@@ -1103,7 +1104,7 @@
   qed
   then obtain K g where "finite K" "disjnt K S"
                and contg: "continuous_on (affine hull T - K) g"
-               and gim:  "g ` (affine hull T - K) \<subseteq> rel_frontier U"
+               and gim:  "g \<in> (affine hull T - K) \<rightarrow> rel_frontier U"
                and gf:   "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
     by (metis aff affine_affine_hull aff_dim_affine_hull
               order_trans [OF \<open>S \<subseteq> T\<close> hull_subset [of T affine]])
@@ -1123,14 +1124,14 @@
 lemma extend_map_affine_to_sphere1:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::topological_space"
   assumes "finite K" "affine U" and contf: "continuous_on (U - K) f"
-      and fim: "f ` (U - K) \<subseteq> T"
+      and fim: "f \<in> (U - K) \<rightarrow> T"
       and comps: "\<And>C. \<lbrakk>C \<in> components(U - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
       and clo: "closedin (top_of_set U) S" and K: "disjnt K S" "K \<subseteq> U"
-  obtains g where "continuous_on (U - L) g" "g ` (U - L) \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+  obtains g where "continuous_on (U - L) g" "g \<in> (U - L) \<rightarrow> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
 proof (cases "K = {}")
   case True
   then show ?thesis
-    by (metis Diff_empty Diff_subset contf fim continuous_on_subset image_subsetI image_eqI subset_iff that)
+    by (metis DiffD1 Diff_empty Diff_subset PiE Pi_I contf continuous_on_subset fim that)
 next
   case False
   have "S \<subseteq> U"
@@ -1194,7 +1195,7 @@
     then obtain r where contr: "continuous_on (U - {a}) r"
                     and rim: "r ` (U - {a}) \<subseteq> sphere a d"  "r ` (U - {a}) \<subseteq> U"
                     and req: "\<And>x. x \<in> sphere a d \<inter> U \<Longrightarrow> r x = x"
-      using \<open>affine U\<close> by (auto simp: retract_of_def retraction_def hull_same)
+      using \<open>affine U\<close> by (force simp: retract_of_def retraction_def hull_same)
     define j where "j \<equiv> \<lambda>x. if x \<in> ball a d then r x else x"
     have kj: "\<And>x. x \<in> S \<Longrightarrow> k (j x) = x"
       using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> j_def subC by auto
@@ -1256,7 +1257,7 @@
       have ST: "\<And>x. x \<in> S \<Longrightarrow> (f \<circ> k \<circ> j) x \<in> T"
       proof (simp add: kj)
         show "\<And>x. x \<in> S \<Longrightarrow> f x \<in> T"
-          using K unfolding disjnt_iff by (metis DiffI \<open>S \<subseteq> U\<close> subsetD fim image_subset_iff)
+          using K \<open>S \<subseteq> U\<close> fT unfolding disjnt_iff by auto
       qed
       moreover have "(f \<circ> k \<circ> j) x \<in> T" if "x \<in> C" "x \<noteq> a" "x \<notin> S" for x
       proof -
@@ -1410,7 +1411,7 @@
       using image_mono order_trans by blast
     moreover have "f ` ((U - L) \<inter> (S \<union> \<Union>G)) \<subseteq> T"
       using fim SUG by blast
-    ultimately show "(\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x) ` (U - L) \<subseteq> T"
+    ultimately show "(\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x) \<in> (U - L) \<rightarrow> T"
        by force
     show "\<And>x. x \<in> S \<Longrightarrow> (if x \<in> S \<union> \<Union>G then f x else g x) = f x"
       by (simp add: F_def G_def)
@@ -1423,15 +1424,15 @@
   assumes "compact S" "convex U" "bounded U" "affine T" "S \<subseteq> T"
       and affTU: "aff_dim T \<le> aff_dim U"
       and contf: "continuous_on S f"
-      and fim: "f ` S \<subseteq> rel_frontier U"
+      and fim: "f \<in> S \<rightarrow> rel_frontier U"
       and ovlap: "\<And>C. C \<in> components(T - S) \<Longrightarrow> C \<inter> L \<noteq> {}"
     obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S"
-                      "continuous_on (T - K) g" "g ` (T - K) \<subseteq> rel_frontier U"
+                      "continuous_on (T - K) g" "g \<in> (T - K) \<rightarrow> rel_frontier U"
                       "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
 proof -
   obtain K g where K: "finite K" "K \<subseteq> T" "disjnt K S"
                and contg: "continuous_on (T - K) g"
-               and gim: "g ` (T - K) \<subseteq> rel_frontier U"
+               and gim: "g \<in> (T - K) \<rightarrow> rel_frontier U"
                and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
      using assms extend_map_affine_to_sphere_cofinite_simple by metis
   have "(\<exists>y C. C \<in> components (T - S) \<and> x \<in> C \<and> y \<in> C \<and> y \<in> L)" if "x \<in> K" for x
@@ -1446,7 +1447,7 @@
   then obtain \<xi> where \<xi>: "\<And>x. x \<in> K \<Longrightarrow> \<exists>C. C \<in> components (T - S) \<and> x \<in> C \<and> \<xi> x \<in> C \<and> \<xi> x \<in> L"
     by metis
   obtain h where conth: "continuous_on (T - \<xi> ` K) h"
-             and him: "h ` (T - \<xi> ` K) \<subseteq> rel_frontier U"
+             and him: "h \<in> (T - \<xi> ` K) \<rightarrow> rel_frontier U"
              and hg: "\<And>x. x \<in> S \<Longrightarrow> h x = g x"
   proof (rule extend_map_affine_to_sphere1 [OF \<open>finite K\<close> \<open>affine T\<close> contg gim, of S "\<xi> ` K"])
     show cloTS: "closedin (top_of_set T) S"
@@ -1475,10 +1476,10 @@
   assumes SUT: "compact S" "convex U" "bounded U" "affine T" "S \<subseteq> T"
       and aff: "aff_dim T \<le> aff_dim U"
       and contf: "continuous_on S f"
-      and fim: "f ` S \<subseteq> rel_frontier U"
+      and fim: "f \<in> S \<rightarrow> rel_frontier U"
       and dis: "\<And>C. \<lbrakk>C \<in> components(T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
  obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
-                   "g ` (T - K) \<subseteq> rel_frontier U"
+                   "g \<in> (T - K) \<rightarrow> rel_frontier U"
                    "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
 proof (cases "S = {}")
   case True
@@ -1615,8 +1616,8 @@
     have "g (closest_point (cbox (- c) c \<inter> T) x) \<in> rel_frontier U"
          if "x \<in> T" "x \<in> K \<longrightarrow> x \<notin> cbox (- b - One) (b + One)" for x
       using gim [THEN subsetD] that cloTK by blast
-    then show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K \<inter> cbox (- (b + One)) (b + One))
-               \<subseteq> rel_frontier U"
+    then show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) \<in> (T - K \<inter> cbox (- (b + One)) (b + One))
+               \<rightarrow> rel_frontier U"
       by force
     show "\<And>x. x \<in> S \<Longrightarrow> (g \<circ> closest_point (cbox (- c) c \<inter> T)) x = f x"
       by simp (metis (mono_tags, lifting) IntI \<open>S \<subseteq> T\<close> cT_ne clo_cT closest_point_refl gf subsetD S_sub_cc)
@@ -1629,23 +1630,26 @@
   assumes SUT: "compact S" "affine T" "S \<subseteq> T"
       and aff: "aff_dim T \<le> DIM('b)" and "0 \<le> r"
       and contf: "continuous_on S f"
-      and fim: "f ` S \<subseteq> sphere a r"
+      and fim: "f \<in> S \<rightarrow> sphere a r"
       and dis: "\<And>C. \<lbrakk>C \<in> components(T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
   obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
-                    "g ` (T - K) \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+                    "g \<in> (T - K) \<rightarrow> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
 proof (cases "r = 0")
   case True
   with fim show ?thesis
     by (rule_tac K="{}" and g = "\<lambda>x. a" in that) (auto)
 next
   case False
-  with assms have "0 < r" by auto
-  then have "aff_dim T \<le> aff_dim (cball a r)"
-    by (simp add: aff aff_dim_cball)
-  then show ?thesis
-    using extend_map_affine_to_sphere_cofinite_gen
-            [OF \<open>compact S\<close> convex_cball bounded_cball \<open>affine T\<close> \<open>S \<subseteq> T\<close> _ contf]
-    using fim by (fastforce simp: assms False that dest: dis)
+  show thesis
+  proof (rule  extend_map_affine_to_sphere_cofinite_gen
+      [OF \<open>compact S\<close> convex_cball bounded_cball \<open>affine T\<close> \<open>S \<subseteq> T\<close> _ contf])
+    have "0 < r"
+      using assms False by auto
+    then show "aff_dim T \<le> aff_dim (cball a r)"
+      by (simp add: aff aff_dim_cball)
+    show "f \<in> S \<rightarrow> rel_frontier (cball a r)"
+      by (simp add: False fim)
+  qed (use dis False that in auto)
 qed
 
 corollary extend_map_UNIV_to_sphere_cofinite:
@@ -1653,10 +1657,10 @@
   assumes "DIM('a) \<le> DIM('b)" and "0 \<le> r"
       and "compact S"
       and "continuous_on S f"
-      and "f ` S \<subseteq> sphere a r"
+      and "f \<in> S \<rightarrow> sphere a r"
       and "\<And>C. \<lbrakk>C \<in> components(- S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
   obtains K g where "finite K" "K \<subseteq> L" "disjnt K S" "continuous_on (- K) g"
-                    "g ` (- K) \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+                    "g \<in> (- K) \<rightarrow> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
   using extend_map_affine_to_sphere_cofinite
         [OF \<open>compact S\<close> affine_UNIV subset_UNIV] assms
   by (metis Compl_eq_Diff_UNIV aff_dim_UNIV of_nat_le_iff)
@@ -1666,9 +1670,9 @@
   assumes aff: "DIM('a) \<le> DIM('b)" and "0 \<le> r"
       and SUT: "compact S"
       and contf: "continuous_on S f"
-      and fim: "f ` S \<subseteq> sphere a r"
+      and fim: "f \<in> S \<rightarrow> sphere a r"
       and dis: "\<And>C. C \<in> components(- S) \<Longrightarrow> \<not> bounded C"
-  obtains g where "continuous_on UNIV g" "g ` UNIV \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+  obtains g where "continuous_on UNIV g" "g \<in> UNIV \<rightarrow> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
   using extend_map_UNIV_to_sphere_cofinite [OF aff \<open>0 \<le> r\<close> \<open>compact S\<close> contf fim, of "{}"]
   by (metis Compl_empty_eq dis subset_empty)
 
@@ -1677,7 +1681,7 @@
   fixes S :: "'a::euclidean_space set"
   assumes "compact S"
     shows "(\<forall>c \<in> components(- S). \<not>bounded c) \<longleftrightarrow>
-           (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1
+           (\<forall>f. continuous_on S f \<and> f \<in> S \<rightarrow> sphere (0::'a) 1
                 \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)))"
        (is "?lhs = ?rhs")
 proof
@@ -1685,14 +1689,14 @@
   show ?rhs
   proof clarify
     fix f :: "'a \<Rightarrow> 'a"
-    assume contf: "continuous_on S f" and fim: "f ` S \<subseteq> sphere 0 1"
-    obtain g where contg: "continuous_on UNIV g" and gim: "range g \<subseteq> sphere 0 1"
+    assume contf: "continuous_on S f" and fim: "f \<in> S \<rightarrow> sphere 0 1"
+    obtain g where contg: "continuous_on UNIV g" and gim: "g \<in> UNIV \<rightarrow> sphere 0 1"
                and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
       by (rule extend_map_UNIV_to_sphere_no_bounded_component [OF _ _ \<open>compact S\<close> contf fim L]) auto
     then obtain c where c: "homotopic_with_canon (\<lambda>h. True) UNIV (sphere 0 1) g (\<lambda>x. c)"
-      using contractible_UNIV nullhomotopic_from_contractible by blast
+      by (metis contractible_UNIV nullhomotopic_from_contractible)
     then show "\<exists>c. homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)"
-      by (metis assms compact_imp_closed contf contg contractible_empty fim gf gim nullhomotopic_from_contractible nullhomotopic_into_sphere_extension)
+      by (metis assms compact_imp_closed contf contg contractible_empty fim gf gim nullhomotopic_from_contractible nullhomotopic_into_sphere_extension image_subset_iff_funcset)
   qed
 next
   assume R [rule_format]: ?rhs
@@ -1717,7 +1721,7 @@
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" and 2: "2 \<le> DIM('a)"
     shows "connected(- S) \<longleftrightarrow>
-           (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1
+           (\<forall>f. continuous_on S f \<and> f \<in> S \<rightarrow> sphere (0::'a) 1
                 \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)))"
        (is "?lhs = ?rhs")
 proof
@@ -2881,7 +2885,7 @@
 corollary homotopy_eqv_simple_connectedness:
   fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
   shows "S homotopy_eqv T \<Longrightarrow> simply_connected S \<longleftrightarrow> simply_connected T"
-  by (simp add: simply_connected_eq_homotopic_circlemaps homotopy_eqv_homotopic_triviality)
+  by (simp add: simply_connected_eq_homotopic_circlemaps homotopy_eqv_homotopic_triviality image_subset_iff_funcset)
 
 
 subsection\<open>Homeomorphism of simple closed curves to circles\<close>
@@ -3673,10 +3677,10 @@
   then obtain g where contg: "continuous_on S g" and f: "\<And>x. x \<in> S \<Longrightarrow> f x = exp(g x)"
     by metis
   obtain a where "homotopic_with_canon (\<lambda>h. True) S (- {of_real 0}) (exp \<circ> g) (\<lambda>x. a)"
-  proof (rule nullhomotopic_through_contractible [OF contg subset_UNIV _ _ contractible_UNIV])
+  proof (rule nullhomotopic_through_contractible [OF contg _ _ _ contractible_UNIV])
     show "continuous_on (UNIV::complex set) exp"
       by (intro continuous_intros)
-    show "range exp \<subseteq> - {0}"
+    show "exp \<in> UNIV \<rightarrow> -{of_real 0}"
       by auto
   qed force
   then have "homotopic_with_canon (\<lambda>h. True) S (- {0}) f (\<lambda>t. a)"
@@ -3719,11 +3723,11 @@
   proof (rule nullhomotopic_through_contractible)
     show "continuous_on S (complex_of_real \<circ> g)"
       by (intro conjI contg continuous_intros)
-    show "(complex_of_real \<circ> g) ` S \<subseteq> \<real>"
+    show "(complex_of_real \<circ> g) \<in> S \<rightarrow> \<real>"
       by auto
     show "continuous_on \<real> (exp \<circ> (*)\<i>)"
       by (intro continuous_intros)
-    show "(exp \<circ> (*)\<i>) ` \<real> \<subseteq> sphere 0 1"
+    show "(exp \<circ> (*)\<i>) \<in> \<real> \<rightarrow> sphere 0 1"
       by (auto simp: complex_is_Real_iff)
   qed (auto simp: convex_Reals convex_imp_contractible)
   moreover have "\<And>x. x \<in> S \<Longrightarrow> (exp \<circ> (*)\<i> \<circ> (complex_of_real \<circ> g)) x = f x"
@@ -4012,7 +4016,7 @@
 proof -
   obtain c where hom: "homotopic_with_canon (\<lambda>h. True) S (-{0}) f (\<lambda>x. c)"
     using nullhomotopic_from_contractible assms
-    by (metis imageE subset_Compl_singleton)
+    by (metis imageE subset_Compl_singleton image_subset_iff_funcset)
   then show ?thesis
     by (metis inessential_eq_continuous_logarithm that)
 qed
@@ -4023,7 +4027,7 @@
       and f: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
   obtains g where "continuous_on S g" "\<And>x. x \<in> S \<Longrightarrow> f x = exp(g x)"
   using covering_space_lift [OF covering_space_exp_punctured_plane S contf]
-  by (metis (full_types) f imageE subset_Compl_singleton)
+  by (metis (full_types) f imageE subset_Compl_singleton image_subset_iff_funcset)
 
 lemma continuous_logarithm_on_cball:
   fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
@@ -4077,7 +4081,7 @@
 lemma inessential_spheremap_2_aux:
   fixes f :: "'a::euclidean_space \<Rightarrow> complex"
   assumes 2: "2 < DIM('a)" and contf: "continuous_on (sphere a r) f" 
-      and fim: "f `(sphere a r) \<subseteq> (sphere 0 1)" 
+      and fim: "f \<in> (sphere a r) \<rightarrow> (sphere 0 1)" 
   obtains c where "homotopic_with_canon (\<lambda>z. True) (sphere a r) (sphere 0 1) f (\<lambda>x. c)"
 proof -
   obtain g where contg: "continuous_on (sphere a r) g" 
@@ -4095,7 +4099,8 @@
     show "continuous_on (sphere a r) (Im \<circ> g)"
       by (intro contg continuous_intros continuous_on_compose)
     show "\<forall>x\<in>sphere a r. f x = exp (\<i> * complex_of_real ((Im \<circ> g) x))"
-      using exp_eq_polar feq fim norm_exp_eq_Re by auto
+      using exp_eq_polar feq fim norm_exp_eq_Re
+      by (auto simp flip: image_subset_iff_funcset)
   qed
   with inessential_eq_continuous_logarithm_circle that show ?thesis 
     by metis
@@ -4104,7 +4109,7 @@
 lemma inessential_spheremap_2:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes a2: "2 < DIM('a)" and b2: "DIM('b) = 2" 
-      and contf: "continuous_on (sphere a r) f" and fim: "f `(sphere a r) \<subseteq> (sphere b s)"
+      and contf: "continuous_on (sphere a r) f" and fim: "f \<in> (sphere a r) \<rightarrow> (sphere b s)"
   obtains c where "homotopic_with_canon (\<lambda>z. True) (sphere a r) (sphere b s) f (\<lambda>x. c)"
 proof (cases "s \<le> 0")
   case True
@@ -4118,20 +4123,20 @@
     by (auto simp: homeomorphic_def)
   then have conth: "continuous_on (sphere b s) h"
        and  contk: "continuous_on (sphere 0 1) k"
-       and  him: "h ` sphere b s \<subseteq> sphere 0 1"
-       and  kim: "k ` sphere 0 1 \<subseteq> sphere b s"
-    by (simp_all add: homeomorphism_def)
+       and  him: "h \<in> sphere b s \<rightarrow> sphere 0 1"
+       and  kim: "k \<in> sphere 0 1 \<rightarrow> sphere b s"
+    by (force simp: homeomorphism_def)+
   obtain c where "homotopic_with_canon (\<lambda>z. True) (sphere a r) (sphere 0 1) (h \<circ> f) (\<lambda>x. c)"
   proof (rule inessential_spheremap_2_aux [OF a2])
     show "continuous_on (sphere a r) (h \<circ> f)"
-      by (meson continuous_on_compose [OF contf] conth continuous_on_subset fim)
-    show "(h \<circ> f) ` sphere a r \<subseteq> sphere 0 1"
+      by (meson contf conth continuous_on_compose continuous_on_subset fim image_subset_iff_funcset)
+    show "(h \<circ> f) \<in> sphere a r \<rightarrow> sphere 0 1"
       using fim him by force
   qed auto
   then have "homotopic_with_canon (\<lambda>f. True) (sphere a r) (sphere b s) (k \<circ> (h \<circ> f)) (k \<circ> (\<lambda>x. c))"
     by (rule homotopic_with_compose_continuous_left [OF _ contk kim])
   moreover have "\<And>x. r = dist a x \<Longrightarrow> f x = k (h (f x))"
-    by (metis fim hk homeomorphism_def image_subset_iff mem_sphere)
+    by (metis fim hk homeomorphism_def image_subset_iff mem_sphere image_subset_iff_funcset)
   ultimately have "homotopic_with_canon (\<lambda>z. True) (sphere a r) (sphere b s) f (\<lambda>x. k c)"
     by (auto intro: homotopic_with_eq)
   then show ?thesis
@@ -4330,16 +4335,16 @@
 
 definition\<^marker>\<open>tag important\<close> Borsukian where
     "Borsukian S \<equiv>
-        \<forall>f. continuous_on S f \<and> f ` S \<subseteq> (- {0::complex})
+        \<forall>f. continuous_on S f \<and> f \<in> S \<rightarrow> (- {0::complex})
             \<longrightarrow> (\<exists>a. homotopic_with_canon (\<lambda>h. True) S (- {0}) f (\<lambda>x. a))"
 
 lemma Borsukian_retraction_gen:
   assumes "Borsukian S" "continuous_on S h" "h ` S = T"
-          "continuous_on T k"  "k ` T \<subseteq> S"  "\<And>y. y \<in> T \<Longrightarrow> h(k y) = y"
+          "continuous_on T k"  "k \<in> T \<rightarrow> S"  "\<And>y. y \<in> T \<Longrightarrow> h(k y) = y"
     shows "Borsukian T"
 proof -
   interpret R: Retracts S h T k
-    using assms by (simp add: Retracts.intro)
+    using assms by (simp add: image_subset_iff_funcset Retracts.intro)
   show ?thesis
     using assms
     apply (clarsimp simp add: Borsukian_def)
@@ -4348,7 +4353,7 @@
 qed
 
 lemma retract_of_Borsukian: "\<lbrakk>Borsukian T; S retract_of T\<rbrakk> \<Longrightarrow> Borsukian S"
-  by (smt (verit) Borsukian_retraction_gen retract_of_def retraction retraction_def retraction_subset)
+  by (smt (verit) Borsukian_retraction_gen retract_of_def retraction retraction_def retraction_subset image_subset_iff_funcset)
 
 lemma homeomorphic_Borsukian: "\<lbrakk>Borsukian S; S homeomorphic T\<rbrakk> \<Longrightarrow> Borsukian T"
   using Borsukian_retraction_gen order_refl
@@ -4374,29 +4379,29 @@
     and T :: "'b::real_normed_vector set"
    assumes "S homotopy_eqv T"
      shows "(Borsukian S \<longleftrightarrow> Borsukian T)"
-  by (meson Borsukian_def assms homotopy_eqv_cohomotopic_triviality_null)
+  by (meson Borsukian_def assms homotopy_eqv_cohomotopic_triviality_null image_subset_iff_funcset)
 
 lemma Borsukian_alt:
   fixes S :: "'a::real_normed_vector set"
   shows
    "Borsukian S \<longleftrightarrow>
-        (\<forall>f g. continuous_on S f \<and> f ` S \<subseteq> -{0} \<and>
-               continuous_on S g \<and> g ` S \<subseteq> -{0}
+        (\<forall>f g. continuous_on S f \<and> f \<in> S \<rightarrow> -{0} \<and>
+               continuous_on S g \<and> g \<in> S \<rightarrow> -{0}
                \<longrightarrow> homotopic_with_canon (\<lambda>h. True) S (- {0::complex}) f g)"
   unfolding Borsukian_def homotopic_triviality
-  by (simp add: path_connected_punctured_universe)
+  by (force simp add: path_connected_punctured_universe)
 
 lemma Borsukian_continuous_logarithm:
   fixes S :: "'a::real_normed_vector set"
   shows "Borsukian S \<longleftrightarrow>
-            (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> (- {0::complex})
+            (\<forall>f. continuous_on S f \<and> f \<in> S \<rightarrow> (- {0::complex})
                  \<longrightarrow> (\<exists>g. continuous_on S g \<and> (\<forall>x \<in> S. f x = exp(g x))))"
   by (simp add: Borsukian_def inessential_eq_continuous_logarithm)
 
 lemma Borsukian_continuous_logarithm_circle:
   fixes S :: "'a::real_normed_vector set"
   shows "Borsukian S \<longleftrightarrow>
-             (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::complex) 1
+             (\<forall>f. continuous_on S f \<and> f \<in> S \<rightarrow> sphere (0::complex) 1
                   \<longrightarrow> (\<exists>g. continuous_on S g \<and> (\<forall>x \<in> S. f x = exp(g x))))"
    (is "?lhs = ?rhs")
 proof
@@ -4405,12 +4410,12 @@
 next
   assume RHS [rule_format]: ?rhs
   show ?lhs
-  proof (clarsimp simp: Borsukian_continuous_logarithm)
+  proof (clarsimp simp: Borsukian_continuous_logarithm Pi_iff)
     fix f :: "'a \<Rightarrow> complex"
-    assume contf: "continuous_on S f" and 0: "0 \<notin> f ` S"
+    assume contf: "continuous_on S f" and 0: "\<forall>i\<in>S. f i \<noteq> 0"
     then have "continuous_on S (\<lambda>x. f x / complex_of_real (cmod (f x)))"
       by (intro continuous_intros) auto
-    moreover have "(\<lambda>x. f x / complex_of_real (cmod (f x))) ` S \<subseteq> sphere 0 1"
+    moreover have "(\<lambda>x. f x / complex_of_real (cmod (f x))) \<in> S \<rightarrow> sphere 0 1"
       using 0 by (auto simp: norm_divide)
     ultimately obtain g where contg: "continuous_on S g"
                   and fg: "\<forall>x \<in> S. f x / complex_of_real (cmod (f x)) = exp(g x)"
@@ -4431,7 +4436,7 @@
 lemma Borsukian_continuous_logarithm_circle_real:
   fixes S :: "'a::real_normed_vector set"
   shows "Borsukian S \<longleftrightarrow>
-         (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::complex) 1
+         (\<forall>f. continuous_on S f \<and> f \<in> S \<rightarrow> sphere (0::complex) 1
               \<longrightarrow> (\<exists>g. continuous_on S (complex_of_real \<circ> g) \<and> (\<forall>x \<in> S. f x = exp(\<i> * of_real(g x)))))"
    (is "?lhs = ?rhs")
 proof
@@ -4439,11 +4444,11 @@
   show ?rhs
   proof (clarify)
     fix f :: "'a \<Rightarrow> complex"
-    assume "continuous_on S f" and f01: "f ` S \<subseteq> sphere 0 1"
+    assume "continuous_on S f" and f01: "f \<in> S \<rightarrow> sphere 0 1"
     then obtain g where contg: "continuous_on S g" and "\<And>x. x \<in> S \<Longrightarrow> f x = exp(g x)"
       using LHS by (auto simp: Borsukian_continuous_logarithm_circle)
     then have "\<forall>x\<in>S. f x = exp (\<i> * complex_of_real ((Im \<circ> g) x))"
-      using f01 exp_eq_polar norm_exp_eq_Re by auto
+      using f01 exp_eq_polar norm_exp_eq_Re by (fastforce simp: Pi_iff)
     then show "\<exists>g. continuous_on S (complex_of_real \<circ> g) \<and> (\<forall>x\<in>S. f x = exp (\<i> * complex_of_real (g x)))"
       by (rule_tac x="Im \<circ> g" in exI) (force intro: continuous_intros contg)
   qed
@@ -4452,7 +4457,7 @@
   show ?lhs
   proof (clarsimp simp: Borsukian_continuous_logarithm_circle)
     fix f :: "'a \<Rightarrow> complex"
-    assume "continuous_on S f" and f01: "f ` S \<subseteq> sphere 0 1"
+    assume "continuous_on S f" and f01: "f \<in> S \<rightarrow> sphere 0 1"
     then obtain g where contg: "continuous_on S (complex_of_real \<circ> g)" and "\<And>x. x \<in> S \<Longrightarrow> f x =  exp(\<i> * of_real(g x))"
       by (metis RHS)
     then show "\<exists>g. continuous_on S g \<and> (\<forall>x\<in>S. f x = exp (g x))"
@@ -4463,17 +4468,18 @@
 lemma Borsukian_circle:
   fixes S :: "'a::real_normed_vector set"
   shows "Borsukian S \<longleftrightarrow>
-         (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::complex) 1
+         (\<forall>f. continuous_on S f \<and> f \<in> S \<rightarrow> sphere (0::complex) 1
               \<longrightarrow> (\<exists>a. homotopic_with_canon (\<lambda>h. True) S (sphere (0::complex) 1) f (\<lambda>x. a)))"
 by (simp add: inessential_eq_continuous_logarithm_circle Borsukian_continuous_logarithm_circle_real)
 
 lemma contractible_imp_Borsukian: "contractible S \<Longrightarrow> Borsukian S"
-  by (meson Borsukian_def nullhomotopic_from_contractible)
+  by (meson Borsukian_def nullhomotopic_from_contractible image_subset_iff_funcset)
 
 lemma simply_connected_imp_Borsukian:
   fixes S :: "'a::real_normed_vector set"
   shows  "\<lbrakk>simply_connected S; locally path_connected S\<rbrakk> \<Longrightarrow> Borsukian S"
-  by (metis (no_types, lifting) Borsukian_continuous_logarithm continuous_logarithm_on_simply_connected image_eqI subset_Compl_singleton)
+  by (smt (verit, del_insts) continuous_logarithm_on_simply_connected Borsukian_continuous_logarithm_circle
+      PiE mem_sphere_0 norm_eq_zero zero_neq_one)
 
 lemma starlike_imp_Borsukian:
   fixes S :: "'a::real_normed_vector set"
@@ -4504,19 +4510,19 @@
                  \<lbrakk>continuous_on S f; continuous_on T g; \<And>x. x \<in> S \<and> x \<in> T \<Longrightarrow> f x = g x\<rbrakk>
            \<Longrightarrow> continuous_on (S \<union> T) (\<lambda>x. if x \<in> S then f x else g x)"
   shows "Borsukian(S \<union> T)"
-proof (clarsimp simp add: Borsukian_continuous_logarithm)
+proof (clarsimp simp add: Borsukian_continuous_logarithm Pi_iff)
   fix f :: "'a \<Rightarrow> complex"
-  assume contf: "continuous_on (S \<union> T) f" and 0: "0 \<notin> f ` (S \<union> T)"
+  assume contf: "continuous_on (S \<union> T) f" and 0: "\<forall>i\<in>S \<union> T. f i \<noteq> 0"
   then have contfS: "continuous_on S f" and contfT: "continuous_on T f"
     using continuous_on_subset by auto
   have "\<lbrakk>continuous_on S f; f ` S \<subseteq> -{0}\<rbrakk> \<Longrightarrow> \<exists>g. continuous_on S g \<and> (\<forall>x \<in> S. f x = exp(g x))"
     using BS by (auto simp: Borsukian_continuous_logarithm)
   then obtain g where contg: "continuous_on S g" and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = exp(g x)"
-    using "0" contfS by blast
+    using "0" contfS by force
   have "\<lbrakk>continuous_on T f; f ` T \<subseteq> -{0}\<rbrakk> \<Longrightarrow> \<exists>g. continuous_on T g \<and> (\<forall>x \<in> T. f x = exp(g x))"
     using BT by (auto simp: Borsukian_continuous_logarithm)
   then obtain h where conth: "continuous_on T h" and fh: "\<And>x. x \<in> T \<Longrightarrow> f x = exp(h x)"
-    using "0" contfT by blast
+    using "0" contfT by force
   show "\<exists>g. continuous_on (S \<union> T) g \<and> (\<forall>x\<in>S \<union> T. f x = exp (g x))"
   proof (cases "S \<inter> T = {}")
     case True
@@ -4592,9 +4598,9 @@
   assumes "Borsukian S" and contf: "continuous_on S f" and fim: "f ` S = T"
       and "compact S" and conn: "\<And>y. y \<in> T \<Longrightarrow> connected {x. x \<in> S \<and> f x = y}"
     shows "Borsukian T"
-proof (clarsimp simp add: Borsukian_continuous_logarithm)
+proof (clarsimp simp: Borsukian_continuous_logarithm Pi_iff)
   fix g :: "'b \<Rightarrow> complex"
-  assume contg: "continuous_on T g" and 0: "0 \<notin> g ` T"
+  assume contg: "continuous_on T g" and 0: "\<forall>i\<in>T. g i \<noteq> 0"
   have "continuous_on S (g \<circ> f)"
     using contf contg continuous_on_compose fim by blast
   moreover have "(g \<circ> f) ` S \<subseteq> -{0}"
@@ -4631,7 +4637,7 @@
     proof (rule continuous_from_closed_graph [of "h ` S"])
       show "compact (h ` S)"
         by (simp add: \<open>compact S\<close> compact_continuous_image conth)
-      show "(h \<circ> f') ` T \<subseteq> h ` S"
+      show "(h \<circ> f') \<in> T \<rightarrow> h ` S"
         by (auto simp: f')
       have "h x = h (f' (f x))" if "x \<in> S" for x
         using * [of "f x"] fim that unfolding constant_on_def by clarsimp (metis f' imageI right_minus_eq)
@@ -4660,10 +4666,10 @@
     shows "Borsukian T"
 proof (clarsimp simp add: Borsukian_continuous_logarithm_circle_real)
   fix g :: "'b \<Rightarrow> complex"
-  assume contg: "continuous_on T g" and gim: "g ` T \<subseteq> sphere 0 1"
+  assume contg: "continuous_on T g" and gim: "g \<in> T \<rightarrow> sphere 0 1"
   have "continuous_on S (g \<circ> f)"
     using contf contg continuous_on_compose fim by blast
-  moreover have "(g \<circ> f) ` S \<subseteq> sphere 0 1"
+  moreover have "(g \<circ> f) \<in> S \<rightarrow> sphere 0 1"
     using fim gim by auto
   ultimately obtain h where cont_cxh: "continuous_on S (complex_of_real \<circ> h)"
                        and gfh: "\<And>x. x \<in> S \<Longrightarrow> (g \<circ> f) x = exp(\<i> * of_real(h x))"
@@ -4677,7 +4683,8 @@
       show "continuous_on {x \<in> S. f x = y} h"
         by (rule continuous_on_subset [OF conth]) auto
       have "compact (S \<inter> f -` {y})"
-        by (rule proper_map_from_compact [OF contf _ \<open>compact S\<close>, of T]) (simp_all add: fim that)
+        using that proper_map_from_compact [OF contf _ \<open>compact S\<close>] fim
+        by force
       then show "compact {x \<in> S. f x = y}" 
         by (auto simp: vimage_def Int_def)
     qed
@@ -4985,7 +4992,7 @@
       with qV qW show ?thesis by force
     qed
     obtain g where contg: "continuous_on U g"
-      and circle: "g ` U \<subseteq> sphere 0 1"
+      and circle: "g \<in> U \<rightarrow> sphere 0 1"
       and S: "\<And>x. x \<in> S \<Longrightarrow> g x = exp(pi * \<i> * q x)"
       and T: "\<And>x. x \<in> T \<Longrightarrow> g x = 1 / exp(pi * \<i> * q x)"
     proof
@@ -5415,7 +5422,7 @@
     proof (rule Borsuk_homotopy_extension_homotopic [OF _ _ continuous_on_const _ homotopic_with_symD [OF a]])
       show "closedin (top_of_set UNIV) S"
         using assms by auto
-      show "range (\<lambda>t. a) \<subseteq> - {0}"
+      show "(\<lambda>t. a) \<in> UNIV \<rightarrow> - {0}"
         using a homotopic_with_imp_subset2 False by blast
     qed (use anr that in \<open>force+\<close>)
     then show ?thesis
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Homeomorphism.thy
--- a/src/HOL/Analysis/Homeomorphism.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Homeomorphism.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -1112,10 +1112,10 @@
   fixes g1 :: "'a \<Rightarrow> 'c::real_normed_vector"
   assumes cov: "covering_space c p S"
       and eq: "g1 a = g2 a"
-      and f: "continuous_on T f"  "f ` T \<subseteq> S"
-      and g1: "continuous_on T g1"  "g1 ` T \<subseteq> c"
+      and f: "continuous_on T f"  "f \<in> T \<rightarrow> S"
+      and g1: "continuous_on T g1"  "g1 \<in> T \<rightarrow> c"
       and fg1: "\<And>x. x \<in> T \<Longrightarrow> f x = p(g1 x)"
-      and g2: "continuous_on T g2"  "g2 ` T \<subseteq> c"
+      and g2: "continuous_on T g2"  "g2 \<in> T \<rightarrow> c"
       and fg2: "\<And>x. x \<in> T \<Longrightarrow> f x = p(g2 x)"
       and u_compt: "U \<in> components T" and "a \<in> U" "x \<in> U"
     shows "g1 x = g2 x"
@@ -1175,9 +1175,9 @@
   fixes g1 :: "'a \<Rightarrow> 'c::real_normed_vector"
   assumes "covering_space c p S"
           "g1 a = g2 a"
-          "continuous_on T f"  "f ` T \<subseteq> S"
-          "continuous_on T g1"  "g1 ` T \<subseteq> c"  "\<And>x. x \<in> T \<Longrightarrow> f x = p(g1 x)"
-          "continuous_on T g2"  "g2 ` T \<subseteq> c"  "\<And>x. x \<in> T \<Longrightarrow> f x = p(g2 x)"
+          "continuous_on T f"  "f \<in> T \<rightarrow> S"
+          "continuous_on T g1"  "g1 \<in> T \<rightarrow> c"  "\<And>x. x \<in> T \<Longrightarrow> f x = p(g1 x)"
+          "continuous_on T g2"  "g2 \<in> T \<rightarrow> c"  "\<And>x. x \<in> T \<Longrightarrow> f x = p(g2 x)"
           "connected T"  "a \<in> T"   "x \<in> T"
    shows "g1 x = g2 x"
   using covering_space_lift_unique_gen [of c p S] in_components_self assms ex_in_conv
@@ -1315,11 +1315,11 @@
     and h :: "real \<times> 'c::real_normed_vector \<Rightarrow> 'b"
   assumes cov: "covering_space C p S"
       and conth: "continuous_on ({0..1} \<times> U) h"
-      and him: "h ` ({0..1} \<times> U) \<subseteq> S"
+      and him: "h \<in> ({0..1} \<times> U) \<rightarrow> S"
       and heq: "\<And>y. y \<in> U \<Longrightarrow> h (0,y) = p(f y)"
-      and contf: "continuous_on U f" and fim: "f ` U \<subseteq> C"
+      and contf: "continuous_on U f" and fim: "f \<in> U \<rightarrow> C"
     obtains k where "continuous_on ({0..1} \<times> U) k"
-                    "k ` ({0..1} \<times> U) \<subseteq> C"
+                    "k \<in> ({0..1} \<times> U) \<rightarrow> C"
                     "\<And>y. y \<in> U \<Longrightarrow> k(0, y) = f y"
                     "\<And>z. z \<in> {0..1} \<times> U \<Longrightarrow> h z = p(k z)"
 proof -
@@ -1459,7 +1459,7 @@
       proof (rule continuous_openin_preimage [OF _ _ opeC'])
         show "continuous_on V (k \<circ> Pair (n/N))"
           by (intro continuous_intros continuous_on_subset [OF contk], auto)
-        show "(k \<circ> Pair (n/N)) ` V \<subseteq> C"
+        show "(k \<circ> Pair (n/N)) \<in> V \<rightarrow> C"
           using kim by (auto simp: \<open>y \<in> V\<close> W)
       qed
       obtain N' where opeUN': "openin (top_of_set U) N'"
@@ -1633,19 +1633,19 @@
           using*V by (simp add: \<open>y \<in> V i\<close> \<open>y \<in> V j\<close> that)
         show conth_y: "continuous_on ({0..1} \<times> {y}) h"
           using VU \<open>y \<in> V j\<close> that by (auto intro: continuous_on_subset [OF conth])
-        show "h ` ({0..1} \<times> {y}) \<subseteq> S"
+        show "h \<in> ({0..1} \<times> {y}) \<rightarrow> S"
           using \<open>y \<in> V i\<close> assms(3) VU that by fastforce
         show "continuous_on ({0..1} \<times> {y}) (fs i)"
           using continuous_on_subset [OF contfs] \<open>i \<in> U\<close>
           by (simp add: \<open>y \<in> V i\<close> subset_iff)
-        show "fs i ` ({0..1} \<times> {y}) \<subseteq> C"
+        show "fs i \<in> ({0..1} \<times> {y}) \<rightarrow> C"
           using "*" \<open>y \<in> V i\<close> \<open>i \<in> U\<close> by fastforce
         show "\<And>x. x \<in> {0..1} \<times> {y} \<Longrightarrow> h x = p (fs i x)"
           using "*" \<open>y \<in> V i\<close> \<open>i \<in> U\<close> by blast
         show "continuous_on ({0..1} \<times> {y}) (fs j)"
           using continuous_on_subset [OF contfs] \<open>j \<in> U\<close>
           by (simp add: \<open>y \<in> V j\<close> subset_iff)
-        show "fs j ` ({0..1} \<times> {y}) \<subseteq> C"
+        show "fs j \<in> ({0..1} \<times> {y}) \<rightarrow> C"
           using "*" \<open>y \<in> V j\<close> \<open>j \<in> U\<close> by fastforce
         show "\<And>x. x \<in> {0..1} \<times> {y} \<Longrightarrow> h x = p (fs j x)"
           using "*" \<open>y \<in> V j\<close> \<open>j \<in> U\<close> by blast
@@ -1660,7 +1660,7 @@
   qed force
   show ?thesis
   proof
-    show "k ` ({0..1} \<times> U) \<subseteq> C"
+    show "k \<in> ({0..1} \<times> U) \<rightarrow> C"
       using V*k VU by fastforce
     show "\<And>y. y \<in> U \<Longrightarrow> k (0, y) = f y"
       by (simp add: V*k)
@@ -1674,11 +1674,11 @@
     and h :: "'c::real_normed_vector \<times> real \<Rightarrow> 'b"
   assumes cov: "covering_space C p S"
       and conth: "continuous_on (U \<times> {0..1}) h"
-      and him: "h ` (U \<times> {0..1}) \<subseteq> S"
+      and him: "h \<in> (U \<times> {0..1}) \<rightarrow> S"
       and heq: "\<And>y. y \<in> U \<Longrightarrow> h (y,0) = p(f y)"
-      and contf: "continuous_on U f" and fim: "f ` U \<subseteq> C"
+      and contf: "continuous_on U f" and fim: "f \<in> U \<rightarrow> C"
   obtains k where "continuous_on (U \<times> {0..1}) k"
-                  "k ` (U \<times> {0..1}) \<subseteq> C"
+                  "k \<in> (U \<times> {0..1}) \<rightarrow> C"
                   "\<And>y. y \<in> U \<Longrightarrow> k(y, 0) = f y"
                   "\<And>z. z \<in> U \<times> {0..1} \<Longrightarrow> h z = p(k z)"
 proof -
@@ -1701,13 +1701,13 @@
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" and g:: "'c::real_normed_vector \<Rightarrow> 'a"
   assumes cov: "covering_space C p S"
       and contg: "continuous_on U g"
-      and gim: "g ` U \<subseteq> C"
+      and gim: "g \<in> U \<rightarrow> C"
       and pgeq: "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
       and hom: "homotopic_with_canon (\<lambda>x. True) U S f f'"
     obtains g' where "continuous_on U g'" "image g' U \<subseteq> C" "\<And>y. y \<in> U \<Longrightarrow> p(g' y) = f' y"
 proof -
   obtain h where conth: "continuous_on ({0..1::real} \<times> U) h"
-             and him: "h ` ({0..1} \<times> U) \<subseteq> S"
+             and him: "h \<in> ({0..1} \<times> U) \<rightarrow> S"
              and h0:  "\<And>x. h(0, x) = f x"
              and h1: "\<And>x. h(1, x) = f' x"
     using hom by (auto simp: homotopic_with_def)
@@ -1717,7 +1717,7 @@
                   and kim: "k ` ({0..1} \<times> U) \<subseteq> C"
                   and k0: "\<And>y. y \<in> U \<Longrightarrow> k(0, y) = g y"
                   and heq: "\<And>z. z \<in> {0..1} \<times> U \<Longrightarrow> h z = p(k z)"
-    using covering_space_lift_homotopy [OF cov conth him _ contg gim] by metis
+    using covering_space_lift_homotopy [OF cov conth him _ contg gim] by (metis image_subset_iff_funcset)
   show ?thesis
   proof
     show "continuous_on U (k \<circ> Pair 1)"
@@ -1743,7 +1743,7 @@
   then obtain b where b: "b \<in> C" "p b = a"
     using covering_space_imp_surjective [OF cov] homotopic_with_imp_subset2 [OF hom]
     by auto
-  then have gim: "(\<lambda>y. b) ` U \<subseteq> C"
+  then have gim: "(\<lambda>y. b) \<in> U \<rightarrow> C"
     by blast
   show ?thesis
   proof (rule covering_space_lift_homotopic_function [OF cov continuous_on_const gim])
@@ -1770,7 +1770,7 @@
   proof (rule covering_space_lift_homotopy [OF cov, of "{undefined}" "g \<circ> fst"])
     show "continuous_on ({0..1::real} \<times> {undefined::'c}) (g \<circ> fst)"
       using \<open>path g\<close> by (intro continuous_intros) (simp add: path_def)
-    show "(g \<circ> fst) ` ({0..1} \<times> {undefined}) \<subseteq> S"
+    show "(g \<circ> fst) \<in> ({0..1} \<times> {undefined}) \<rightarrow> S"
       using pag by (auto simp: path_image_def)
     show "(g \<circ> fst) (0, y) = p a" if "y \<in> {undefined}" for y::'c
       by (metis comp_def fst_conv pas pathstart_def)
@@ -1815,13 +1815,13 @@
 proof -
   obtain h :: "real \<times> real \<Rightarrow> 'b"
      where conth: "continuous_on ({0..1} \<times> {0..1}) h"
-       and him: "h ` ({0..1} \<times> {0..1}) \<subseteq> S"
+       and him: "h \<in> ({0..1} \<times> {0..1}) \<rightarrow> S"
        and h0: "\<And>x. h (0, x) = g1 x" and h1: "\<And>x. h (1, x) = g2 x"
        and heq0: "\<And>t. t \<in> {0..1} \<Longrightarrow> h (t, 0) = g1 0"
        and heq1: "\<And>t. t \<in> {0..1} \<Longrightarrow> h (t, 1) = g1 1"
-    using hom by (auto simp: homotopic_paths_def homotopic_with_def pathstart_def pathfinish_def)
+    using hom by (auto simp: homotopic_paths_def homotopic_with_def pathstart_def pathfinish_def image_subset_iff_funcset)
   obtain k where contk: "continuous_on ({0..1} \<times> {0..1}) k"
-             and kim: "k ` ({0..1} \<times> {0..1}) \<subseteq> C"
+             and kim: "k \<in> ({0..1} \<times> {0..1}) \<rightarrow> C"
              and kh2: "\<And>y. y \<in> {0..1} \<Longrightarrow> k (y, 0) = h2 0"
              and hpk: "\<And>z. z \<in> {0..1} \<times> {0..1} \<Longrightarrow> h z = p (k z)"
   proof (rule covering_space_lift_homotopy_alt [OF cov conth him])
@@ -1830,11 +1830,11 @@
   qed (use path_image_def pih2 in \<open>fastforce+\<close>)
   have contg1: "continuous_on {0..1} g1" and contg2: "continuous_on {0..1} g2"
     using \<open>path g1\<close> \<open>path g2\<close> path_def by blast+
-  have g1im: "g1 ` {0..1} \<subseteq> S" and g2im: "g2 ` {0..1} \<subseteq> S"
+  have g1im: "g1 \<in> {0..1} \<rightarrow> S" and g2im: "g2 \<in> {0..1} \<rightarrow> S"
     using path_image_def pig1 pig2 by auto
   have conth1: "continuous_on {0..1} h1" and conth2: "continuous_on {0..1} h2"
     using \<open>path h1\<close> \<open>path h2\<close> path_def by blast+
-  have h1im: "h1 ` {0..1} \<subseteq> C" and h2im: "h2 ` {0..1} \<subseteq> C"
+  have h1im: "h1 \<in> {0..1} \<rightarrow> C" and h2im: "h2 \<in> {0..1} \<rightarrow> C"
     using path_image_def pih1 pih2 by auto
   show ?thesis
     unfolding homotopic_paths pathstart_def pathfinish_def
@@ -1911,13 +1911,13 @@
     and f :: "'c::real_normed_vector \<Rightarrow> 'b"
   assumes cov: "covering_space C p S" and "a \<in> C" "z \<in> U"
       and U: "path_connected U" "locally path_connected U"
-      and contf: "continuous_on U f" and fim: "f ` U \<subseteq> S"
+      and contf: "continuous_on U f" and fim: "f \<in> U \<rightarrow> S"
       and feq: "f z = p a"
       and hom: "\<And>r. \<lbrakk>path r; path_image r \<subseteq> U; pathstart r = z; pathfinish r = z\<rbrakk>
                      \<Longrightarrow> \<exists>q. path q \<and> path_image q \<subseteq> C \<and>
                              pathstart q = a \<and> pathfinish q = a \<and>
                              homotopic_paths S (f \<circ> r) (p \<circ> q)"
-  obtains g where "continuous_on U g" "g ` U \<subseteq> C" "g z = a" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
+  obtains g where "continuous_on U g" "g \<in> U \<rightarrow> C" "g z = a" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
 proof -
   have *: "\<exists>g h. path g \<and> path_image g \<subseteq> U \<and>
                  pathstart g = z \<and> pathfinish g = y \<and>
@@ -1934,7 +1934,7 @@
       show "path (f \<circ> g)"
         using \<open>path g\<close> \<open>path_image g \<subseteq> U\<close> contf continuous_on_subset path_continuous_image by blast
       show "path_image (f \<circ> g) \<subseteq> S"
-        by (metis \<open>path_image g \<subseteq> U\<close> fim image_mono path_image_compose subset_trans)
+        by (metis \<open>path_image g \<subseteq> U\<close> fim image_mono path_image_compose subset_trans image_subset_iff_funcset)
       show "pathstart (f \<circ> g) = p a"
         by (simp add: feq pastg pathstart_compose)
     qed auto
@@ -1975,11 +1975,11 @@
           show "g ` (*\<^sub>R) 2 ` {0..1/2} \<subseteq> U"
             using g path_image_def by fastforce
         qed auto
-        show "(f \<circ> g \<circ> (*\<^sub>R) 2) ` {0..1/2} \<subseteq> S"
-          using g(2) path_image_def fim by fastforce
-        show "(h \<circ> (*\<^sub>R) 2) ` {0..1/2} \<subseteq> C"
+        show "(f \<circ> g \<circ> (*\<^sub>R) 2) \<in> {0..1/2} \<rightarrow> S"
+          using g(2) fim by (fastforce simp: path_image_def image_subset_iff_funcset)
+        show "(h \<circ> (*\<^sub>R) 2) \<in> {0..1/2} \<rightarrow> C"
           using h path_image_def by fastforce
-        show "q' ` {0..1/2} \<subseteq> C"
+        show "q' \<in> {0..1/2} \<rightarrow> C"
           using \<open>path_image q' \<subseteq> C\<close> path_image_def by fastforce
         show "\<And>x. x \<in> {0..1/2} \<Longrightarrow> (f \<circ> g \<circ> (*\<^sub>R) 2) x = p (q' x)"
           by (auto simp: joinpaths_def pq'_eq)
@@ -2000,11 +2000,11 @@
           show "reversepath g' ` (\<lambda>t. 2 *\<^sub>R t - 1) ` {1/2<..1} \<subseteq> U"
             using g' by (auto simp: path_image_def reversepath_def)
         qed (use g' in auto)
-        show "(f \<circ> reversepath g' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)) ` {1/2<..1} \<subseteq> S"
+        show "(f \<circ> reversepath g' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)) \<in> {1/2<..1} \<rightarrow> S"
           using g'(2) path_image_def fim by (auto simp: image_subset_iff path_image_def reversepath_def)
-        show "q' ` {1/2<..1} \<subseteq> C"
+        show "q' \<in> {1/2<..1} \<rightarrow> C"
           using \<open>path_image q' \<subseteq> C\<close> path_image_def by fastforce
-        show "(reversepath h' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)) ` {1/2<..1} \<subseteq> C"
+        show "(reversepath h' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)) \<in> {1/2<..1} \<rightarrow> C"
           using h' by (simp add: path_image_def reversepath_def subset_eq)
         show "\<And>x. x \<in> {1/2<..1} \<Longrightarrow> (f \<circ> reversepath g' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)) x = p (q' x)"
           by (auto simp: joinpaths_def pq'_eq)
@@ -2036,7 +2036,7 @@
       using*[OF \<open>y \<in> U\<close>]  by (metis l atLeastAtMost_iff order_refl pathfinish_def zero_le_one)
     show "l z = a"
       using l [of "linepath z z" z "linepath a a"] by (auto simp: assms)
-    show LC: "l ` U \<subseteq> C"
+    show LC: "l \<in> U \<rightarrow> C"
       by (clarify dest!: *) (metis (full_types) l pathfinish_in_path_image subsetCE)
     have "\<exists>T. openin (top_of_set U) T \<and> y \<in> T \<and> T \<subseteq> U \<inter> l -` X"
          if X: "openin (top_of_set C) X" and "y \<in> U" "l y \<in> X" for X y
@@ -2080,7 +2080,8 @@
         have "openin (top_of_set S) (W \<inter> p' -` (W' \<inter> X))"
         proof (rule openin_trans)
           show "openin (top_of_set W) (W \<inter> p' -` (W' \<inter> X))"
-            using X \<open>W' \<subseteq> C\<close> by (intro continuous_openin_preimage [OF contp' p'im]) (auto simp: openin_open)
+            using X \<open>W' \<subseteq> C\<close>
+            by (metis continuous_on_open contp' homUW' homeomorphism_image2 inf.assoc inf.orderE openin_open)
           show "openin (top_of_set S) W"
             using WV by blast
         qed
@@ -2154,6 +2155,7 @@
       qed
     qed
     then show "continuous_on U l"
+      using vimage_eq openin_subopen continuous_on_open_gen [OF LC]
       by (metis IntD1 IntD2 vimage_eq openin_subopen continuous_on_open_gen [OF LC])
   qed
 qed
@@ -2163,11 +2165,11 @@
     and f :: "'c::real_normed_vector \<Rightarrow> 'b"
   assumes cov: "covering_space C p S" "a \<in> C" "z \<in> U"
       and U: "path_connected U" "locally path_connected U"
-      and contf: "continuous_on U f" and fim: "f ` U \<subseteq> S"
+      and contf: "continuous_on U f" and fim: "f \<in> U \<rightarrow> S"
       and feq: "f z = p a"
       and hom: "\<And>r. \<lbrakk>path r; path_image r \<subseteq> U; pathstart r = z; pathfinish r = z\<rbrakk>
                      \<Longrightarrow> \<exists>b. homotopic_paths S (f \<circ> r) (linepath b b)"
-  obtains g where "continuous_on U g" "g ` U \<subseteq> C" "g z = a" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
+  obtains g where "continuous_on U g" "g \<in> U \<rightarrow> C" "g z = a" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
 proof (rule covering_space_lift_general [OF cov U contf fim feq])
   fix r
   assume "path r" "path_image r \<subseteq> U" "pathstart r = z" "pathfinish r = z"
@@ -2190,9 +2192,9 @@
     and f :: "'c::real_normed_vector \<Rightarrow> 'b"
   assumes cov: "covering_space C p S" "a \<in> C" "z \<in> U"
       and scU: "simply_connected U" and lpcU: "locally path_connected U"
-      and contf: "continuous_on U f" and fim: "f ` U \<subseteq> S"
+      and contf: "continuous_on U f" and fim: "f \<in> U \<rightarrow> S"
       and feq: "f z = p a"
-  obtains g where "continuous_on U g" "g ` U \<subseteq> C" "g z = a" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
+  obtains g where "continuous_on U g" "g \<in> U \<rightarrow> C" "g z = a" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
 proof (rule covering_space_lift_stronger [OF cov _ lpcU contf fim feq])
   show "path_connected U"
     using scU simply_connected_eq_contractible_loop_some by blast
@@ -2211,8 +2213,8 @@
     and f :: "'c::real_normed_vector \<Rightarrow> 'b"
   assumes cov: "covering_space C p S"
       and U: "simply_connected U"  "locally path_connected U"
-      and contf: "continuous_on U f" and fim: "f ` U \<subseteq> S"
-    obtains g where "continuous_on U g" "g ` U \<subseteq> C" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
+      and contf: "continuous_on U f" and fim: "f \<in> U \<rightarrow> S"
+    obtains g where "continuous_on U g" "g \<in> U \<rightarrow> C" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
 proof (cases "U = {}")
   case True
   with that show ?thesis by auto
@@ -2220,7 +2222,7 @@
   case False
   then obtain z where "z \<in> U" by blast
   then obtain a where "a \<in> C" "f z = p a"
-    by (metis cov covering_space_imp_surjective fim image_iff image_subset_iff)
+    by (metis cov covering_space_imp_surjective fim image_iff Pi_iff)
   then show ?thesis
     by (metis that covering_space_lift_strong [OF cov _ \<open>z \<in> U\<close> U contf fim])
 qed
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Homotopy.thy
--- a/src/HOL/Analysis/Homotopy.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Homotopy.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -124,6 +124,14 @@
      "homotopic_with_canon P X Y f g \<Longrightarrow> g ` X \<subseteq> Y"
   by (meson continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)
 
+lemma homotopic_with_imp_funspace1:
+     "homotopic_with_canon P X Y f g \<Longrightarrow> f \<in> X \<rightarrow> Y"
+  using homotopic_with_imp_subset1 by blast
+
+lemma homotopic_with_imp_funspace2:
+     "homotopic_with_canon P X Y f g \<Longrightarrow> g \<in> X \<rightarrow> Y"
+  using homotopic_with_imp_subset2 by blast
+
 lemma homotopic_with_subset_left:
      "\<lbrakk>homotopic_with_canon P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with_canon P Z Y f g"
   unfolding homotopic_with_def by (auto elim!: continuous_on_subset ex_forward)
@@ -150,7 +158,7 @@
   have *: "continuous_map (prod_topology ?I01 X) (prod_topology ?I01 X) ?j"
   proof -
     have "continuous_map (prod_topology ?I01 X) (subtopology (prod_topology euclideanreal X) ({0..1} \<times> topspace X)) ?j"
-      by (simp add: continuous_map_into_subtopology [OF 1] image_subset_iff)
+      by (simp add: continuous_map_into_subtopology [OF 1] image_subset_iff flip: image_subset_iff_funcset)
     then show ?thesis
       by (simp add: prod_topology_subtopology(1))
   qed
@@ -262,14 +270,14 @@
   by (metis assms homotopic_with_compose_continuous_map_left homotopic_with_compose_continuous_map_right homotopic_with_imp_continuous_maps homotopic_with_trans)
 
 proposition homotopic_with_compose_continuous_right:
-    "\<lbrakk>homotopic_with_canon (\<lambda>f. p (f \<circ> h)) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
+    "\<lbrakk>homotopic_with_canon (\<lambda>f. p (f \<circ> h)) X Y f g; continuous_on W h; h \<in> W \<rightarrow> X\<rbrakk>
      \<Longrightarrow> homotopic_with_canon p W Y (f \<circ> h) (g \<circ> h)"
-  by (simp add: homotopic_with_compose_continuous_map_right)
+  by (simp add: homotopic_with_compose_continuous_map_right image_subset_iff_funcset)
 
 proposition homotopic_with_compose_continuous_left:
-     "\<lbrakk>homotopic_with_canon (\<lambda>f. p (h \<circ> f)) X Y f g; continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
+     "\<lbrakk>homotopic_with_canon (\<lambda>f. p (h \<circ> f)) X Y f g; continuous_on Y h; h \<in> Y \<rightarrow> Z\<rbrakk>
       \<Longrightarrow> homotopic_with_canon p X Z (h \<circ> f) (h \<circ> g)"
-  by (simp add: homotopic_with_compose_continuous_map_left)
+  by (simp add: homotopic_with_compose_continuous_map_left image_subset_iff_funcset)
 
 lemma homotopic_from_subtopology:
    "homotopic_with P X X' f g \<Longrightarrow> homotopic_with P (subtopology X S) X' f g"
@@ -397,15 +405,15 @@
 
 text\<open>Homotopic triviality implicitly incorporates path-connectedness.\<close>
 lemma homotopic_triviality:
-  shows  "(\<forall>f g. continuous_on S f \<and> f ` S \<subseteq> T \<and>
-                 continuous_on S g \<and> g ` S \<subseteq> T
+  shows  "(\<forall>f g. continuous_on S f \<and> f \<in> S \<rightarrow> T \<and>
+                 continuous_on S g \<and> g \<in> S \<rightarrow> T
                  \<longrightarrow> homotopic_with_canon (\<lambda>x. True) S T f g) \<longleftrightarrow>
           (S = {} \<or> path_connected T) \<and>
-          (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> T \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c)))"
+          (\<forall>f. continuous_on S f \<and> f \<in> S \<rightarrow> T \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c)))"
           (is "?lhs = ?rhs")
 proof (cases "S = {} \<or> T = {}")
   case True then show ?thesis
-    by (auto simp: homotopic_on_emptyI)
+    by (auto simp: homotopic_on_emptyI simp flip: image_subset_iff_funcset)
 next
   case False show ?thesis
   proof
@@ -418,7 +426,7 @@
         using False homotopic_constant_maps [of "top_of_set S" "top_of_set T" a b] by auto
     qed
     moreover
-    have "\<exists>c. homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c)" if "continuous_on S f" "f ` S \<subseteq> T" for f
+    have "\<exists>c. homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c)" if "continuous_on S f" "f \<in> S \<rightarrow> T" for f
       using False LHS continuous_on_const that by blast
     ultimately show ?rhs
       by (simp add: path_connected_component)
@@ -429,9 +437,9 @@
     show ?lhs
     proof clarify
       fix f g
-      assume "continuous_on S f" "f ` S \<subseteq> T" "continuous_on S g" "g ` S \<subseteq> T"
+      assume "continuous_on S f" "f \<in> S \<rightarrow> T" "continuous_on S g" "g \<in> S \<rightarrow> T"
       obtain c d where c: "homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c)" and d: "homotopic_with_canon (\<lambda>x. True) S T g (\<lambda>x. d)"
-        using False \<open>continuous_on S f\<close> \<open>f ` S \<subseteq> T\<close>  RHS \<open>continuous_on S g\<close> \<open>g ` S \<subseteq> T\<close> by blast
+        using RHS \<open>continuous_on S f\<close> \<open>continuous_on S g\<close> \<open>f \<in> S \<rightarrow> T\<close> \<open>g \<in> S \<rightarrow> T\<close> by presburger
       with T have "path_component T c d"
         by (metis False ex_in_conv homotopic_with_imp_subset2 image_subset_iff path_connected_component)
       then have "homotopic_with_canon (\<lambda>x. True) S T (\<lambda>x. c) (\<lambda>x. d)"
@@ -454,7 +462,7 @@
 lemma homotopic_paths:
    "homotopic_paths S p q \<longleftrightarrow>
       (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and>
-          h ` ({0..1} \<times> {0..1}) \<subseteq> S \<and>
+          h \<in> ({0..1} \<times> {0..1}) \<rightarrow> S \<and>
           (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
           (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
           (\<forall>t \<in> {0..1::real}. pathstart(h \<circ> Pair t) = pathstart p \<and>
@@ -500,7 +508,7 @@
   assumes "path p"
       and pips: "path_image p \<subseteq> S"
       and contf: "continuous_on {0..1} f"
-      and f01:"f ` {0..1} \<subseteq> {0..1}"
+      and f01 :"f \<in> {0..1} \<rightarrow> {0..1}"
       and [simp]: "f(0) = 0" "f(1) = 1"
       and q: "\<And>t. t \<in> {0..1} \<Longrightarrow> q(t) = p(f t)"
     shows "homotopic_paths S p q"
@@ -508,15 +516,15 @@
   have contp: "continuous_on {0..1} p"
     by (metis \<open>path p\<close> path_def)
   then have "continuous_on {0..1} (p \<circ> f)"
-    using contf continuous_on_compose continuous_on_subset f01 by blast
+    by (meson assms(4) contf continuous_on_compose continuous_on_subset image_subset_iff_funcset)
   then have "path q"
     by (simp add: path_def) (metis q continuous_on_cong)
   have piqs: "path_image q \<subseteq> S"
-    by (metis (no_types, opaque_lifting) pips f01 image_subset_iff path_image_def q)
+    by (smt (verit, ccfv_threshold) Pi_iff assms(2) assms(4) assms(7) image_subset_iff path_defs(4))
   have fb0: "\<And>a b. \<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> (1 - a) * f b + a * b"
     using f01 by force
   have fb1: "\<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> (1 - a) * f b + a * b \<le> 1" for a b
-    using f01 [THEN subsetD, of "f b"] by (simp add: convex_bound_le)
+    by (intro convex_bound_le) (use f01 in auto)
   have "homotopic_paths S q p"
   proof (rule homotopic_paths_trans)
     show "homotopic_paths S q (p \<circ> f)"
@@ -585,9 +593,9 @@
   done
 
 proposition homotopic_paths_continuous_image:
-    "\<lbrakk>homotopic_paths S f g; continuous_on S h; h ` S \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t (h \<circ> f) (h \<circ> g)"
+    "\<lbrakk>homotopic_paths S f g; continuous_on S h; h \<in> S \<rightarrow> t\<rbrakk> \<Longrightarrow> homotopic_paths t (h \<circ> f) (h \<circ> g)"
   unfolding homotopic_paths_def
-  by (simp add: homotopic_with_compose_continuous_map_left pathfinish_compose pathstart_compose)
+  by (simp add: homotopic_with_compose_continuous_map_left pathfinish_compose pathstart_compose image_subset_iff_funcset)
 
 
 subsection\<open>Group properties for homotopy of paths\<close>
@@ -711,9 +719,9 @@
   by (auto intro: homotopic_with_eq [OF homotopic_with_refl [where f = p, THEN iffD2]])
 
 proposition homotopic_loops_continuous_image:
-   "\<lbrakk>homotopic_loops S f g; continuous_on S h; h ` S \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t (h \<circ> f) (h \<circ> g)"
+   "\<lbrakk>homotopic_loops S f g; continuous_on S h; h \<in> S \<rightarrow> t\<rbrakk> \<Longrightarrow> homotopic_loops t (h \<circ> f) (h \<circ> g)"
   unfolding homotopic_loops_def
-  by (simp add: homotopic_with_compose_continuous_map_left pathfinish_def pathstart_def)
+  by (simp add: homotopic_with_compose_continuous_map_left pathfinish_def pathstart_def image_subset_iff_funcset)
 
 
 subsection\<open>Relations between the two variants of homotopy\<close>
@@ -731,11 +739,11 @@
   have pip: "path_image p \<subseteq> S" by (metis assms homotopic_loops_imp_subset)
   let ?A = "{0..1::real} \<times> {0..1::real}"
   obtain h where conth: "continuous_on ?A h"
-             and hs: "h ` ?A \<subseteq> S"
+             and hs: "h \<in> ?A \<rightarrow> S"
              and h0[simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(0,x) = p x"
              and h1[simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(1,x) = a"
              and ends: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish (h \<circ> Pair t) = pathstart (h \<circ> Pair t)"
-    using assms by (auto simp: homotopic_loops homotopic_with)
+    using assms by (auto simp: homotopic_loops homotopic_with image_subset_iff_funcset)
   have conth0: "path (\<lambda>u. h (u, 0))"
     unfolding path_def
   proof (rule continuous_on_compose [of _ _ h, unfolded o_def])
@@ -783,12 +791,13 @@
                +++ subpath (fst y) 0 (\<lambda>u. h (u, 0))) (snd y)" 
     have "continuous_on ?A ?h"
       by (intro continuous_on_homotopic_join_lemma; simp add: path_defs joinpaths_def subpath_def conth c1 c2)
-    moreover have "?h ` ?A \<subseteq> S"
+    moreover have "?h \<in> ?A \<rightarrow> S"
+      using hs
       unfolding joinpaths_def subpath_def
-      by (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le)
+      by (force simp: algebra_simps mult_le_one mult_left_le  intro: adhoc_le)
   ultimately show "continuous_map (prod_topology (top_of_set {0..1}) (top_of_set {0..1}))
                          (top_of_set S) ?h"
-    by (simp add: subpath_reversepath)
+    by (simp add: subpath_reversepath image_subset_iff_funcset)
   qed (use ploop in \<open>simp_all add: reversepath_def path_defs joinpaths_def o_def subpath_def conth c1 c2\<close>)
   moreover have "homotopic_paths S ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))
                                    (linepath (pathstart p) (pathstart p))"
@@ -999,7 +1008,7 @@
       show "continuous_on {0..1} ?f"
         unfolding split_01
         by (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def dest!: t2)+
-      show "?f ` {0..1} \<subseteq> {0..1}"
+      show "?f \<in> {0..1} \<rightarrow> {0..1}"
         using False assms
         by (force simp: field_simps not_le mult_left_mono affine_ineq dest!: 1 2)
       show "(subpath u v g +++ subpath v w g) t = subpath u w g (?f t)" if "t \<in> {0..1}" for t 
@@ -1057,12 +1066,12 @@
   assumes "path_component S a b"
   shows "homotopic_loops S (linepath a a) (linepath b b)"
 proof -
-  obtain g :: "real \<Rightarrow> 'a" where g: "continuous_on {0..1} g" "g ` {0..1} \<subseteq> S" "g 0 = a" "g 1 = b"
+  obtain g :: "real \<Rightarrow> 'a" where g: "continuous_on {0..1} g" "g \<in> {0..1} \<rightarrow> S" "g 0 = a" "g 1 = b"
     using assms by (auto simp: path_defs)
   then have "continuous_on ({0..1} \<times> {0..1}) (g \<circ> fst)"
     by (fastforce intro!: continuous_intros)+
   with g show ?thesis
-    by (auto simp: homotopic_loops_def homotopic_with_def path_defs image_subset_iff)
+    by (auto simp: homotopic_loops_def homotopic_with_def path_defs Pi_iff)
 qed
 
 lemma homotopic_loops_imp_path_component_value:
@@ -1289,39 +1298,39 @@
 
 lemma nullhomotopic_through_contractible:
   fixes S :: "_::topological_space set"
-  assumes f: "continuous_on S f" "f ` S \<subseteq> T"
-      and g: "continuous_on T g" "g ` T \<subseteq> U"
+  assumes f: "continuous_on S f" "f \<in> S \<rightarrow> T"
+      and g: "continuous_on T g" "g \<in> T \<rightarrow> U"
       and T: "contractible T"
     obtains c where "homotopic_with_canon (\<lambda>h. True) S U (g \<circ> f) (\<lambda>x. c)"
 proof -
   obtain b where b: "homotopic_with_canon (\<lambda>x. True) T T id (\<lambda>x. b)"
     using assms by (force simp: contractible_def)
   have "homotopic_with_canon (\<lambda>f. True) T U (g \<circ> id) (g \<circ> (\<lambda>x. b))"
-    by (metis Abstract_Topology.continuous_map_subtopology_eu b g homotopic_with_compose_continuous_map_left)
+    by (metis b continuous_map_subtopology_eu g homotopic_with_compose_continuous_map_left image_subset_iff_funcset)
   then have "homotopic_with_canon (\<lambda>f. True) S U (g \<circ> id \<circ> f) (g \<circ> (\<lambda>x. b) \<circ> f)"
-    by (simp add: f homotopic_with_compose_continuous_map_right)
+    by (simp add: f homotopic_with_compose_continuous_map_right image_subset_iff_funcset)
   then show ?thesis
     by (simp add: comp_def that)
 qed
 
 lemma nullhomotopic_into_contractible:
-  assumes f: "continuous_on S f" "f ` S \<subseteq> T"
+  assumes f: "continuous_on S f" "f \<in> S \<rightarrow> T"
       and T: "contractible T"
     obtains c where "homotopic_with_canon (\<lambda>h. True) S T f (\<lambda>x. c)"
   by (rule nullhomotopic_through_contractible [OF f, of id T]) (use assms in auto)
 
 lemma nullhomotopic_from_contractible:
-  assumes f: "continuous_on S f" "f ` S \<subseteq> T"
+  assumes f: "continuous_on S f" "f \<in> S \<rightarrow> T"
       and S: "contractible S"
     obtains c where "homotopic_with_canon (\<lambda>h. True) S T f (\<lambda>x. c)"
   by (auto simp: comp_def intro: nullhomotopic_through_contractible [OF continuous_on_id _ f S])
 
 lemma homotopic_through_contractible:
   fixes S :: "_::real_normed_vector set"
-  assumes "continuous_on S f1" "f1 ` S \<subseteq> T"
-          "continuous_on T g1" "g1 ` T \<subseteq> U"
-          "continuous_on S f2" "f2 ` S \<subseteq> T"
-          "continuous_on T g2" "g2 ` T \<subseteq> U"
+  assumes "continuous_on S f1" "f1 \<in> S \<rightarrow> T"
+          "continuous_on T g1" "g1 \<in> T \<rightarrow> U"
+          "continuous_on S f2" "f2 \<in> S \<rightarrow> T"
+          "continuous_on T g2" "g2 \<in> T \<rightarrow> U"
           "contractible T" "path_connected U"
    shows "homotopic_with_canon (\<lambda>h. True) S U (g1 \<circ> f1) (g2 \<circ> f2)"
 proof -
@@ -1346,8 +1355,8 @@
 
 lemma homotopic_into_contractible:
   fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
-  assumes f: "continuous_on S f" "f ` S \<subseteq> T"
-    and g: "continuous_on S g" "g ` S \<subseteq> T"
+  assumes f: "continuous_on S f" "f \<in> S \<rightarrow> T"
+    and g: "continuous_on S g" "g \<in> S \<rightarrow> T"
     and T: "contractible T"
   shows "homotopic_with_canon (\<lambda>h. True) S T f g"
   using homotopic_through_contractible [of S f T id T g id]
@@ -1355,8 +1364,8 @@
 
 lemma homotopic_from_contractible:
   fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
-  assumes f: "continuous_on S f" "f ` S \<subseteq> T"
-    and g: "continuous_on S g" "g ` S \<subseteq> T"
+  assumes f: "continuous_on S f" "f \<in> S \<rightarrow> T"
+    and g: "continuous_on S g" "g \<in> S \<rightarrow> T"
     and "contractible S" "path_connected T"
   shows "homotopic_with_canon (\<lambda>h. True) S T f g"
   using homotopic_through_contractible [of S id S f T id g]
@@ -1478,15 +1487,15 @@
   shows "contractible (S \<times> T)"
 proof -
   obtain a h where conth: "continuous_on ({0..1} \<times> S) h"
-             and hsub: "h ` ({0..1} \<times> S) \<subseteq> S"
+             and hsub: "h \<in> ({0..1} \<times> S) \<rightarrow> S"
              and [simp]: "\<And>x. x \<in> S \<Longrightarrow> h (0, x) = x"
              and [simp]: "\<And>x. x \<in> S \<Longrightarrow>  h (1::real, x) = a"
-    using S by (auto simp: contractible_def homotopic_with)
+    using S by (force simp: contractible_def homotopic_with)
   obtain b k where contk: "continuous_on ({0..1} \<times> T) k"
-             and ksub: "k ` ({0..1} \<times> T) \<subseteq> T"
+             and ksub: "k \<in> ({0..1} \<times> T) \<rightarrow> T"
              and [simp]: "\<And>x. x \<in> T \<Longrightarrow> k (0, x) = x"
              and [simp]: "\<And>x. x \<in> T \<Longrightarrow>  k (1::real, x) = b"
-    using T by (auto simp: contractible_def homotopic_with)
+    using T by (force simp: contractible_def homotopic_with)
   show ?thesis
     apply (simp add: contractible_def homotopic_with)
     apply (rule exI [where x=a])
@@ -3210,7 +3219,7 @@
     by (rule hom)
   then have "homotopic_with_canon Q U t (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
     apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
-    using Q by (auto simp: conth imh)
+    using Q conth imh by force+
   then show ?thesis
   proof (rule homotopic_with_eq; simp)
     show "\<And>h k. (\<And>x. x \<in> U \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
@@ -3240,7 +3249,7 @@
     by (metis hom)
   then have "homotopic_with_canon Q U t (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
     apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
-    using Q by (auto simp: conth imh)
+    using Q conth imh by force+
   then have "homotopic_with_canon Q U t f (\<lambda>x. h c)"
   proof (rule homotopic_with_eq)
     show "\<And>x. x \<in> topspace (top_of_set U) \<Longrightarrow> f x = (h \<circ> (k \<circ> f)) x"
@@ -3281,7 +3290,7 @@
     by (rule hom)
   then have "homotopic_with_canon Q t U (f \<circ> h \<circ> k) (g \<circ> h \<circ> k)"
     apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
-    using Q by (auto simp: contk imk)
+    using Q contk imk by force+
   then show ?thesis
   proof (rule homotopic_with_eq)
     show "f x = (f \<circ> h \<circ> k) x" "g x = (g \<circ> h \<circ> k) x" 
@@ -3312,7 +3321,7 @@
   proof (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
     show "\<And>h. \<lbrakk>continuous_map (top_of_set S) (top_of_set U) h; P h\<rbrakk> \<Longrightarrow> Q (h \<circ> k)"
       using Q by auto
-  qed (auto simp: contk imk)
+  qed (use contk imk in force)+
   moreover have "homotopic_with_canon Q t U f (\<lambda>x. c)"
     using homotopic_with_eq [OF \<section>] feq Qeq by fastforce
   ultimately show ?thesis 
@@ -3789,15 +3798,10 @@
              (infix "homotopy'_eqv" 50)
   where "S homotopy_eqv T \<equiv> top_of_set S homotopy_equivalent_space top_of_set T"
 
-
-
-
-
 lemma homeomorphic_imp_homotopy_eqv: "S homeomorphic T \<Longrightarrow> S homotopy_eqv T"
   unfolding homeomorphic_def homeomorphism_def homotopy_equivalent_space_def
   by (metis continuous_map_subtopology_eu homotopic_with_id2 openin_imp_subset openin_subtopology_self topspace_euclidean_subtopology)
 
-
 lemma homotopy_eqv_inj_linear_image:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f"
@@ -3814,25 +3818,25 @@
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
-      and f: "continuous_on U f" "f ` U \<subseteq> T"
-      and g: "continuous_on U g" "g ` U \<subseteq> T"
-      and homUS: "\<And>f g. \<lbrakk>continuous_on U f; f ` U \<subseteq> S;
-                         continuous_on U g; g ` U \<subseteq> S\<rbrakk>
+      and f: "continuous_on U f" "f \<in> U \<rightarrow> T"
+      and g: "continuous_on U g" "g \<in> U \<rightarrow> T"
+      and homUS: "\<And>f g. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> S;
+                         continuous_on U g; g \<in> U \<rightarrow> S\<rbrakk>
                          \<Longrightarrow> homotopic_with_canon (\<lambda>x. True) U S f g"
     shows "homotopic_with_canon (\<lambda>x. True) U T f g"
 proof -
-  obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
-               and k: "continuous_on T k" "k ` T \<subseteq> S"
+  obtain h k where h: "continuous_on S h" "h \<in> S \<rightarrow> T"
+               and k: "continuous_on T k" "k \<in> T \<rightarrow> S"
                and hom: "homotopic_with_canon (\<lambda>x. True) S S (k \<circ> h) id"
                         "homotopic_with_canon (\<lambda>x. True) T T (h \<circ> k) id"
-    using assms by (auto simp: homotopy_equivalent_space_def)
+    using assms by (force simp: homotopy_equivalent_space_def image_subset_iff_funcset)
   have "homotopic_with_canon (\<lambda>f. True) U S (k \<circ> f) (k \<circ> g)"
   proof (rule homUS)
     show "continuous_on U (k \<circ> f)" "continuous_on U (k \<circ> g)"
-      using continuous_on_compose continuous_on_subset f g k by blast+
+      using continuous_on_compose continuous_on_subset f g k by (metis funcset_image)+
   qed (use f g k in \<open>(force simp: o_def)+\<close> )
   then have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
-    by (rule homotopic_with_compose_continuous_left; simp add: h)
+    by (simp add: h homotopic_with_compose_continuous_map_left image_subset_iff_funcset)
   moreover have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> k \<circ> f) (id \<circ> f)"
     by (rule homotopic_with_compose_continuous_right [where X=T and Y=T]; simp add: hom f)
   moreover have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> k \<circ> g) (id \<circ> g)"
@@ -3847,11 +3851,11 @@
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
-    shows "(\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> S \<and>
-                   continuous_on U g \<and> g ` U \<subseteq> S
+    shows "(\<forall>f g. continuous_on U f \<and> f \<in> U \<rightarrow> S \<and>
+                   continuous_on U g \<and> g \<in> U \<rightarrow> S
                    \<longrightarrow> homotopic_with_canon (\<lambda>x. True) U S f g) \<longleftrightarrow>
-           (\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> T \<and>
-                  continuous_on U g \<and> g ` U \<subseteq> T
+           (\<forall>f g. continuous_on U f \<and> f \<in> U \<rightarrow> T \<and>
+                  continuous_on U g \<and> g \<in> U \<rightarrow> T
                   \<longrightarrow> homotopic_with_canon (\<lambda>x. True) U T f g)"
       (is "?lhs = ?rhs")
 proof
@@ -3873,20 +3877,20 @@
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
-      and f: "continuous_on T f" "f ` T \<subseteq> U"
-      and homSU: "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> U\<rbrakk>
+      and f: "continuous_on T f" "f \<in> T \<rightarrow> U"
+      and homSU: "\<And>f. \<lbrakk>continuous_on S f; f \<in> S \<rightarrow> U\<rbrakk>
                       \<Longrightarrow> \<exists>c. homotopic_with_canon (\<lambda>x. True) S U f (\<lambda>x. c)"
   obtains c where "homotopic_with_canon (\<lambda>x. True) T U f (\<lambda>x. c)"
 proof -
-  obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
-               and k: "continuous_on T k" "k ` T \<subseteq> S"
+  obtain h k where h: "continuous_on S h" "h \<in> S \<rightarrow> T"
+               and k: "continuous_on T k" "k \<in> T \<rightarrow> S"
                and hom: "homotopic_with_canon (\<lambda>x. True) S S (k \<circ> h) id"
                         "homotopic_with_canon (\<lambda>x. True) T T (h \<circ> k) id"
-    using assms by (auto simp: homotopy_equivalent_space_def)
+    using assms by (force simp: homotopy_equivalent_space_def image_subset_iff_funcset)
   obtain c where "homotopic_with_canon (\<lambda>x. True) S U (f \<circ> h) (\<lambda>x. c)"
   proof (rule exE [OF homSU])
     show "continuous_on S (f \<circ> h)"
-      using continuous_on_compose continuous_on_subset f h by blast
+      by (metis continuous_on_compose continuous_on_subset f h funcset_image)
   qed (use f h in force)
   then have "homotopic_with_canon (\<lambda>x. True) T U ((f \<circ> h) \<circ> k) ((\<lambda>x. c) \<circ> k)"
     by (rule homotopic_with_compose_continuous_right [where X=S]) (use k in auto)
@@ -3902,9 +3906,9 @@
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
-    shows "(\<forall>f. continuous_on S f \<and> f ` S \<subseteq> U
+    shows "(\<forall>f. continuous_on S f \<and> f \<in> S \<rightarrow> U
                 \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) S U f (\<lambda>x. c))) \<longleftrightarrow>
-           (\<forall>f. continuous_on T f \<and> f ` T \<subseteq> U
+           (\<forall>f. continuous_on T f \<and> f \<in> T \<rightarrow> U
                 \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) T U f (\<lambda>x. c)))"
 by (rule iffI; metis assms homotopy_eqv_cohomotopic_triviality_null_imp homotopy_equivalent_space_sym)
 
@@ -3914,20 +3918,20 @@
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
-      and f: "continuous_on U f" "f ` U \<subseteq> T"
-      and homSU: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S\<rbrakk>
+      and f: "continuous_on U f" "f \<in> U \<rightarrow> T"
+      and homSU: "\<And>f. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> S\<rbrakk>
                       \<Longrightarrow> \<exists>c. homotopic_with_canon (\<lambda>x. True) U S f (\<lambda>x. c)"
     shows "\<exists>c. homotopic_with_canon (\<lambda>x. True) U T f (\<lambda>x. c)"
 proof -
-  obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
-               and k: "continuous_on T k" "k ` T \<subseteq> S"
+  obtain h k where h: "continuous_on S h" "h \<in> S \<rightarrow> T"
+               and k: "continuous_on T k" "k \<in> T \<rightarrow> S"
                and hom: "homotopic_with_canon (\<lambda>x. True) S S (k \<circ> h) id"
                         "homotopic_with_canon (\<lambda>x. True) T T (h \<circ> k) id"
-    using assms by (auto simp: homotopy_equivalent_space_def)
+    using assms by (force simp: homotopy_equivalent_space_def image_subset_iff_funcset)
   obtain c::'a where "homotopic_with_canon (\<lambda>x. True) U S (k \<circ> f) (\<lambda>x. c)"
   proof (rule exE [OF homSU [of "k \<circ> f"]])
     show "continuous_on U (k \<circ> f)"
-      using continuous_on_compose continuous_on_subset f k by blast
+      using continuous_on_compose continuous_on_subset f k by (metis funcset_image)
   qed (use f k in force)
   then have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
     by (rule homotopic_with_compose_continuous_left [where Y=S]) (use h in auto)
@@ -3943,9 +3947,9 @@
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
-    shows "(\<forall>f. continuous_on U f \<and> f ` U \<subseteq> S
+    shows "(\<forall>f. continuous_on U f \<and> f \<in> U \<rightarrow> S
                   \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) U S f (\<lambda>x. c))) \<longleftrightarrow>
-           (\<forall>f. continuous_on U f \<and> f ` U \<subseteq> T
+           (\<forall>f. continuous_on U f \<and> f \<in> U \<rightarrow> T
                   \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) U T f (\<lambda>x. c)))"
 by (rule iffI; metis assms homotopy_eqv_homotopic_triviality_null_imp homotopy_equivalent_space_sym)
 
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Lindelof_Spaces.thy
--- a/src/HOL/Analysis/Lindelof_Spaces.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Lindelof_Spaces.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -253,7 +253,7 @@
         show "x \<in> \<Union> (\<Union> (\<V> ` I))" if "x \<in> topspace X" for x
         proof -
           have "f x \<in> topspace Y"
-            by (meson f image_subset_iff proper_map_imp_subset_topspace that)
+            using f proper_map_imp_subset_topspace that by fastforce
           then show ?thesis
             using that I by auto
         qed
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Lipschitz.thy
--- a/src/HOL/Analysis/Lipschitz.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Lipschitz.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -125,8 +125,8 @@
 proof
   show "?lhs \<Longrightarrow> ?rhs"
     by (force simp add: Lipschitz_continuous_map_pos lipschitz_on_def less_le_not_le)
-  show "?rhs \<Longrightarrow> ?lhs"
-  by (metis Lipschitz_continuous_map_def lipschitz_onD mdist_euclidean_metric mspace_euclidean_metric top_greatest)
+  show "?rhs \<Longrightarrow> ?lhs"                             
+    by (auto simp: Lipschitz_continuous_map_def lipschitz_on_def)
 qed
 
 subsubsection \<open>Continuity\<close>
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Path_Connected.thy
--- a/src/HOL/Analysis/Path_Connected.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Path_Connected.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -1484,7 +1484,7 @@
 
 lemma path_connectedin_iff_path_connected_real [simp]:
      "path_connectedin euclideanreal S \<longleftrightarrow> path_connected S"
-  by (simp add: path_connectedin path_connected_def path_defs)
+  by (simp add: path_connectedin path_connected_def path_defs image_subset_iff_funcset) 
 
 lemma path_connected_component: "path_connected S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S. path_component S x y)"
   unfolding path_connected_def path_component_def by auto
@@ -1929,12 +1929,12 @@
 definition path_components_of :: "'a topology \<Rightarrow> 'a set set"
   where "path_components_of X \<equiv> path_component_of_set X ` topspace X"
 
-lemma pathin_canon_iff: "pathin (top_of_set T) g \<longleftrightarrow> path g \<and> g ` {0..1} \<subseteq> T"
-  by (simp add: path_def pathin_def)
+lemma pathin_canon_iff: "pathin (top_of_set T) g \<longleftrightarrow> path g \<and> g \<in> {0..1} \<rightarrow> T"
+  by (simp add: path_def pathin_def image_subset_iff_funcset)
 
 lemma path_component_of_canon_iff [simp]:
   "path_component_of (top_of_set T) a b \<longleftrightarrow> path_component T a b"
-  by (simp add: path_component_of_def pathin_canon_iff path_defs)
+  by (simp add: path_component_of_def pathin_canon_iff path_defs image_subset_iff_funcset)
 
 lemma path_component_in_topspace:
    "path_component_of X x y \<Longrightarrow> x \<in> topspace X \<and> y \<in> topspace X"
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Retracts.thy
--- a/src/HOL/Analysis/Retracts.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Retracts.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -100,7 +100,7 @@
   proof (simp add: retraction_def retract_of_def, intro exI conjI)
     show "continuous_on U (g \<circ> h')"
       by (meson continuous_on_compose continuous_on_subset h' hom homeomorphism_cont1)
-    show "(g \<circ> h') ` U \<subseteq> S'"
+    show "(g \<circ> h') \<in> U \<rightarrow> S'"
       using h'  by clarsimp (metis hom subsetD homeomorphism_def imageI)
     show "\<forall>x\<in>S'. (g \<circ> h') x = x"
       by clarsimp (metis h'h hom homeomorphism_def)
@@ -136,7 +136,7 @@
   proof (simp add: retraction_def retract_of_def, intro exI conjI)
     show "continuous_on U (g \<circ> h')"
       by (meson continuous_on_compose continuous_on_subset h' hom homeomorphism_cont1)
-    show "(g \<circ> h') ` U \<subseteq> T"
+    show "(g \<circ> h') \<in> U \<rightarrow> T"
       using h'  by clarsimp (metis hom subsetD homeomorphism_def imageI)
     show "\<forall>x\<in>T. (g \<circ> h') x = x"
       by clarsimp (metis h'h hom homeomorphism_def)
@@ -173,11 +173,11 @@
 
 lemma ANR_imp_absolute_neighbourhood_extensor:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "ANR S" and contf: "continuous_on T f" and "f ` T \<subseteq> S"
+  assumes "ANR S" and contf: "continuous_on T f" and "f \<in> T \<rightarrow> S"
       and cloUT: "closedin (top_of_set U) T"
   obtains V g where "T \<subseteq> V" "openin (top_of_set U) V"
                     "continuous_on V g"
-                    "g ` V \<subseteq> S" "\<And>x. x \<in> T \<Longrightarrow> g x = f x"
+                    "g \<in> V \<rightarrow> S" "\<And>x. x \<in> T \<Longrightarrow> g x = f x"
 proof -
   have "aff_dim S < int (DIM('b \<times> real))"
     using aff_dim_le_DIM [of S] by simp
@@ -194,7 +194,7 @@
   obtain g h where homgh: "homeomorphism S S' g h"
     using hom by (force simp: homeomorphic_def)
   have "continuous_on (f ` T) g"
-    by (meson \<open>f ` T \<subseteq> S\<close> continuous_on_subset homeomorphism_def homgh)
+    by (metis PiE assms(3) continuous_on_subset homeomorphism_cont1 homgh image_subset_iff)
   then have contgf: "continuous_on T (g \<circ> f)"
     by (intro continuous_on_compose contf)
   have gfTC: "(g \<circ> f) ` T \<subseteq> C"
@@ -202,7 +202,7 @@
     have "g ` S = S'"
       by (metis (no_types) homeomorphism_def homgh)
     then show ?thesis
-      by (metis (no_types) assms(3) cloCS closedin_def image_comp image_mono order.trans topspace_euclidean_subtopology)
+      by (metis PiE assms(3) cloCS closedin_def image_comp image_mono image_subset_iff order.trans topspace_euclidean_subtopology)
   qed
   obtain f' where contf': "continuous_on U f'"
               and "f' ` U \<subseteq> C"
@@ -211,10 +211,10 @@
   show ?thesis
   proof (rule_tac V = "U \<inter> f' -` D" and g = "h \<circ> r \<circ> f'" in that)
     show "T \<subseteq> U \<inter> f' -` D"
-      using cloUT closedin_imp_subset \<open>S' \<subseteq> D\<close> \<open>f ` T \<subseteq> S\<close> eq homeomorphism_image1 homgh
+      using cloUT closedin_imp_subset \<open>S' \<subseteq> D\<close> \<open>f \<in> T \<rightarrow> S\<close> eq homeomorphism_image1 homgh
       by fastforce
     show ope: "openin (top_of_set U) (U \<inter> f' -` D)"
-      using  \<open>f' ` U \<subseteq> C\<close> by (auto simp: opD contf' continuous_openin_preimage)
+      by (meson \<open>f' ` U \<subseteq> C\<close> contf' continuous_openin_preimage image_subset_iff_funcset opD)
     have conth: "continuous_on (r ` f' ` (U \<inter> f' -` D)) h"
     proof (rule continuous_on_subset [of S'])
       show "continuous_on S' h"
@@ -222,12 +222,12 @@
     qed (use \<open>r ` D \<subseteq> S'\<close> in blast)
     show "continuous_on (U \<inter> f' -` D) (h \<circ> r \<circ> f')"
       by (blast intro: continuous_on_compose conth continuous_on_subset [OF contr] continuous_on_subset [OF contf'])
-    show "(h \<circ> r \<circ> f') ` (U \<inter> f' -` D) \<subseteq> S"
+    show "(h \<circ> r \<circ> f') \<in> (U \<inter> f' -` D) \<rightarrow> S"
       using \<open>homeomorphism S S' g h\<close>  \<open>f' ` U \<subseteq> C\<close>  \<open>r ` D \<subseteq> S'\<close>
       by (auto simp: homeomorphism_def)
     show "\<And>x. x \<in> T \<Longrightarrow> (h \<circ> r \<circ> f') x = f x"
-      using \<open>homeomorphism S S' g h\<close> \<open>f ` T \<subseteq> S\<close> eq
-      by (auto simp: rid homeomorphism_def)
+      using \<open>homeomorphism S S' g h\<close> \<open>f \<in> T \<rightarrow> S\<close> eq
+      by (metis PiE comp_apply homeomorphism_def image_iff rid)
   qed
 qed
 
@@ -240,7 +240,7 @@
 proof -
   obtain g h where hom: "homeomorphism S S' g h"
     using assms by (force simp: homeomorphic_def)
-  obtain h: "continuous_on S' h" " h ` S' \<subseteq> S"
+  obtain h: "continuous_on S' h" " h \<in> S' \<rightarrow> S"
     using hom homeomorphism_def by blast
     from ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> h clo]
   obtain V h' where "S' \<subseteq> V" and opUV: "openin (top_of_set U) V"
@@ -251,7 +251,7 @@
   proof (simp add: retraction_def retract_of_def, intro exI conjI \<open>S' \<subseteq> V\<close>)
     show "continuous_on V (g \<circ> h')"
       by (meson continuous_on_compose continuous_on_subset h'(1) h'(2) hom homeomorphism_cont1)
-    show "(g \<circ> h') ` V \<subseteq> S'"
+    show "(g \<circ> h') \<in> V \<rightarrow> S'"
       using h'  by clarsimp (metis hom subsetD homeomorphism_def imageI)
     show "\<forall>x\<in>S'. (g \<circ> h') x = x"
       by clarsimp (metis h'h hom homeomorphism_def)
@@ -269,29 +269,29 @@
 
 corollary neighbourhood_extension_into_ANR:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> T" and "ANR T" "closed S"
+  assumes contf: "continuous_on S f" and fim: "f \<in> S \<rightarrow> T" and "ANR T" "closed S"
   obtains V g where "S \<subseteq> V" "open V" "continuous_on V g"
-                    "g ` V \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+                    "g \<in> V \<rightarrow> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
   using ANR_imp_absolute_neighbourhood_extensor [OF  \<open>ANR T\<close> contf fim]
   by (metis \<open>closed S\<close> closed_closedin open_openin subtopology_UNIV)
 
 lemma absolute_neighbourhood_extensor_imp_ANR:
   fixes S :: "'a::euclidean_space set"
   assumes "\<And>f :: 'a * real \<Rightarrow> 'a.
-           \<And>U T. \<lbrakk>continuous_on T f;  f ` T \<subseteq> S;
+           \<And>U T. \<lbrakk>continuous_on T f;  f \<in> T \<rightarrow> S;
                   closedin (top_of_set U) T\<rbrakk>
                  \<Longrightarrow> \<exists>V g. T \<subseteq> V \<and> openin (top_of_set U) V \<and>
-                       continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x \<in> T. g x = f x)"
+                       continuous_on V g \<and> g \<in> V \<rightarrow> S \<and> (\<forall>x \<in> T. g x = f x)"
   shows "ANR S"
 proof (clarsimp simp: ANR_def)
   fix U and T :: "('a * real) set"
   assume "S homeomorphic T" and clo: "closedin (top_of_set U) T"
   then obtain g h where hom: "homeomorphism S T g h"
     by (force simp: homeomorphic_def)
-  obtain h: "continuous_on T h" " h ` T \<subseteq> S"
+  obtain h: "continuous_on T h" " h \<in> T \<rightarrow> S"
     using hom homeomorphism_def by blast
   obtain V h' where "T \<subseteq> V" and opV: "openin (top_of_set U) V"
-                and h': "continuous_on V h'" "h' ` V \<subseteq> S"
+                and h': "continuous_on V h'" "h' \<in> V \<rightarrow> S"
               and h'h: "\<forall>x\<in>T. h' x = h x"
     using assms [OF h clo] by blast
   have [simp]: "T \<subseteq> U"
@@ -299,9 +299,9 @@
   have "T retract_of V"
   proof (simp add: retraction_def retract_of_def, intro exI conjI \<open>T \<subseteq> V\<close>)
     show "continuous_on V (g \<circ> h')"
-      by (meson continuous_on_compose continuous_on_subset h' hom homeomorphism_cont1)
-    show "(g \<circ> h') ` V \<subseteq> T"
-      using h'  by clarsimp (metis hom subsetD homeomorphism_def imageI)
+      by (meson continuous_on_compose continuous_on_subset h' hom homeomorphism_def image_subset_iff_funcset)
+    show "(g \<circ> h') \<in> V \<rightarrow> T"
+      using h' hom homeomorphism_image1 by fastforce
     show "\<forall>x\<in>T. (g \<circ> h') x = x"
       by clarsimp (metis h'h hom homeomorphism_def)
   qed
@@ -313,10 +313,10 @@
   fixes S :: "'a::euclidean_space set"
   shows "ANR S \<longleftrightarrow>
          (\<forall>f :: 'a * real \<Rightarrow> 'a.
-          \<forall>U T. continuous_on T f \<longrightarrow> f ` T \<subseteq> S \<longrightarrow>
+          \<forall>U T. continuous_on T f \<longrightarrow> f \<in> T \<rightarrow> S \<longrightarrow>
                 closedin (top_of_set U) T \<longrightarrow>
                (\<exists>V g. T \<subseteq> V \<and> openin (top_of_set U) V \<and>
-                       continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x \<in> T. g x = f x)))" (is "_ = ?rhs")
+                       continuous_on V g \<and> g \<in> V \<rightarrow> S \<and> (\<forall>x \<in> T. g x = f x)))" (is "_ = ?rhs")
 proof
   assume "ANR S" then show ?rhs
     by (metis ANR_imp_absolute_neighbourhood_extensor)
@@ -431,7 +431,7 @@
           by (metis \<open>retraction X S r\<close> image_mono image_subset_iff_subset_vimage inf_le2 retraction)
       qed
     qed
-    show "(f \<circ> r \<circ> h) ` (W \<inter> h -` X) \<subseteq> S'"
+    show "(f \<circ> r \<circ> h) \<in> (W \<inter> h -` X) \<rightarrow> S'"
       using \<open>retraction X S r\<close> hom
       by (auto simp: retraction_def homeomorphism_def)
     show "\<forall>x\<in>S'. (f \<circ> r \<circ> h) x = x"
@@ -529,15 +529,15 @@
     by (auto simp: contractible_def homotopic_with_def)
   then have "a \<in> S"
     by (metis all_not_in_conv atLeastAtMost_iff image_subset_iff mem_Sigma_iff order_refl zero_le_one)
-  have "\<exists>g. continuous_on W g \<and> g ` W \<subseteq> S \<and> (\<forall>x\<in>T. g x = f x)"
-         if      f: "continuous_on T f" "f ` T \<subseteq> S"
+  have "\<exists>g. continuous_on W g \<and> g \<in> W \<rightarrow> S \<and> (\<forall>x\<in>T. g x = f x)"
+         if      f: "continuous_on T f" "f \<in> T \<rightarrow> S"
             and WT: "closedin (top_of_set W) T"
          for W T and f :: "'a \<times> real \<Rightarrow> 'a"
   proof -
     obtain U g
       where "T \<subseteq> U" and WU: "openin (top_of_set W) U"
         and contg: "continuous_on U g"
-        and "g ` U \<subseteq> S" and gf: "\<And>x. x \<in> T \<Longrightarrow> g x = f x"
+        and "g \<in> U \<rightarrow> S" and gf: "\<And>x. x \<in> T \<Longrightarrow> g x = f x"
       using iffD1 [OF ANR_eq_absolute_neighbourhood_extensor \<open>ANR S\<close>, rule_format, OF f WT]
       by auto
     have WWU: "closedin (top_of_set W) (W - U)"
@@ -570,7 +570,7 @@
         show "continuous_on (W - V') g"
           by (metis Diff_subset_conv \<open>W - U \<subseteq> V'\<close> contg continuous_on_subset Un_commute)
         show "(\<lambda>x. (j x, g x)) ` (W - V') \<subseteq> {0..1} \<times> S"
-          using j \<open>g ` U \<subseteq> S\<close> \<open>W - U \<subseteq> V'\<close> by fastforce
+          using j \<open>g \<in> U \<rightarrow> S\<close> \<open>W - U \<subseteq> V'\<close> by fastforce
       qed
       show "continuous_on W (\<lambda>x. if x \<in> W - V then a else h (j x, g x))"
       proof (subst Weq, rule continuous_on_cases_local)
@@ -583,12 +583,12 @@
         have "j(x, y) \<in> {0..1}"
           using j that by blast
         moreover have "g(x, y) \<in> S"
-          using \<open>V' \<inter> V = {}\<close> \<open>W - U \<subseteq> V'\<close> \<open>g ` U \<subseteq> S\<close> that by fastforce
+          using \<open>V' \<inter> V = {}\<close> \<open>W - U \<subseteq> V'\<close> \<open>g \<in> U \<rightarrow> S\<close> that by fastforce
         ultimately show ?thesis
           using hS by blast
       qed
-      with \<open>a \<in> S\<close> \<open>g ` U \<subseteq> S\<close>
-      show "(\<lambda>x. if x \<in> W - V then a else h (j x, g x)) ` W \<subseteq> S"
+      with \<open>a \<in> S\<close> \<open>g \<in> U \<rightarrow> S\<close>
+      show "(\<lambda>x. if x \<in> W - V then a else h (j x, g x)) \<in> W \<rightarrow> S"
         by auto
     next
       show "\<forall>x\<in>T. (if x \<in> W - V then a else h (j x, g x)) = f x"
@@ -596,7 +596,7 @@
     qed
   qed
   then show ?lhs
-    by (simp add: AR_eq_absolute_extensor)
+    by (simp add: AR_eq_absolute_extensor image_subset_iff_funcset)
 qed
 
 
@@ -606,17 +606,17 @@
   shows "ANR S"
 proof (clarsimp simp add: ANR_eq_absolute_neighbourhood_extensor)
   fix f::"'a \<times> real \<Rightarrow> 'a" and U W
-  assume W: "continuous_on W f" "f ` W \<subseteq> S" "closedin (top_of_set U) W"
-  then obtain r where "S \<subseteq> T" and r: "continuous_on T r" "r ` T \<subseteq> S" "\<forall>x\<in>S. r x = x" "continuous_on W f" "f ` W \<subseteq> S"
+  assume W: "continuous_on W f" "f \<in> W \<rightarrow> S" "closedin (top_of_set U) W"
+  then obtain r where "S \<subseteq> T" and r: "continuous_on T r" "r \<in> T \<rightarrow> S" "\<forall>x\<in>S. r x = x" "continuous_on W f" "f \<in> W \<rightarrow> S"
                                      "closedin (top_of_set U) W"
-    by (meson ST retract_of_def retraction_def)
+    by (metis ST retract_of_def retraction_def)
   then have "f ` W \<subseteq> T"
     by blast
-  with W obtain V g where V: "W \<subseteq> V" "openin (top_of_set U) V" "continuous_on V g" "g ` V \<subseteq> T" "\<forall>x\<in>W. g x = f x"
-    by (metis ANR_imp_absolute_neighbourhood_extensor \<open>ANR T\<close>)
-  with r have "continuous_on V (r \<circ> g) \<and> (r \<circ> g) ` V \<subseteq> S \<and> (\<forall>x\<in>W. (r \<circ> g) x = f x)"
-    by (metis (no_types, lifting) comp_apply continuous_on_compose continuous_on_subset image_subset_iff)
-  then show "\<exists>V. W \<subseteq> V \<and> openin (top_of_set U) V \<and> (\<exists>g. continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x\<in>W. g x = f x))"
+  with W obtain V g where V: "W \<subseteq> V" "openin (top_of_set U) V" "continuous_on V g" "g \<in> V \<rightarrow> T" "\<forall>x\<in>W. g x = f x"
+    by (smt (verit) ANR_imp_absolute_neighbourhood_extensor Pi_I assms(1) funcset_mem image_subset_iff_funcset)
+  with r have "continuous_on V (r \<circ> g) \<and> (r \<circ> g) \<in> V \<rightarrow> S \<and> (\<forall>x\<in>W. (r \<circ> g) x = f x)"
+    by (smt (verit, del_insts) Pi_iff comp_apply continuous_on_compose continuous_on_subset image_subset_iff_funcset)
+  then show "\<exists>V. W \<subseteq> V \<and> openin (top_of_set U) V \<and> (\<exists>g. continuous_on V g \<and> g \<in> V \<rightarrow> S \<and> (\<forall>x\<in>W. g x = f x))"
     by (meson V)
 qed
 
@@ -775,7 +775,8 @@
   show ?thesis
     apply (simp add: retract_of_def retraction_def \<open>S \<subseteq> U\<close> \<open>T \<subseteq> U\<close>)
     apply (rule_tac x="\<lambda>x. if x \<in> S' then g x else h x" in exI)
-    using ST UST \<open>S \<subseteq> S'\<close> \<open>S' \<inter> T' = W\<close> \<open>T \<subseteq> T'\<close> cont geqr heqr r_def by auto
+    using ST UST \<open>S \<subseteq> S'\<close> \<open>S' \<inter> T' = W\<close> \<open>T \<subseteq> T'\<close> cont geqr heqr r_def
+    by (smt (verit, del_insts) IntI Pi_I Un_iff image_subset_iff r0 subsetD) 
 qed
 
 
@@ -883,14 +884,14 @@
   have "W0 \<subseteq> U"
     using \<open>W \<subseteq> U\<close> cloWW0 closedin_subset by fastforce
   obtain r0
-    where "S \<inter> T \<subseteq> W0" and contr0: "continuous_on W0 r0" and "r0 ` W0 \<subseteq> S \<inter> T"
+    where "S \<inter> T \<subseteq> W0" and contr0: "continuous_on W0 r0" and "r0 \<in> W0 \<rightarrow> S \<inter> T"
       and r0 [simp]: "\<And>x. x \<in> S \<inter> T \<Longrightarrow> r0 x = x"
     using ret  by (force simp: retract_of_def retraction_def)
   have ST: "x \<in> W \<Longrightarrow> x \<in> S \<longleftrightarrow> x \<in> T" for x
     using assms by (auto simp: W_def setdist_sing_in_set dest!: setdist_eq_0_closedin)
   define r where "r \<equiv> \<lambda>x. if x \<in> W0 then r0 x else x"
   have "r ` (W0 \<union> S) \<subseteq> S" "r ` (W0 \<union> T) \<subseteq> T"
-    using \<open>r0 ` W0 \<subseteq> S \<inter> T\<close> r_def by auto
+    using \<open>r0 \<in> W0 \<rightarrow> S \<inter> T\<close> r_def by auto
   have contr: "continuous_on (W0 \<union> (S \<union> T)) r"
   unfolding r_def
   proof (rule continuous_on_cases_local [OF _ _ contr0 continuous_on_id])
@@ -907,21 +908,21 @@
               closedin_Un closedin_imp_subset closedin_trans)
   obtain W1 g where "W0 \<union> S \<subseteq> W1" and contg: "continuous_on W1 g"
                 and opeSW1: "openin (top_of_set S') W1"
-                and "g ` W1 \<subseteq> S" and geqr: "\<And>x. x \<in> W0 \<union> S \<Longrightarrow> g x = r x"
-  proof (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> _ \<open>r ` (W0 \<union> S) \<subseteq> S\<close> cloS'WS])
+                and "g \<in> W1 \<rightarrow> S" and geqr: "\<And>x. x \<in> W0 \<union> S \<Longrightarrow> g x = r x"
+  proof (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> _ _ cloS'WS])
     show "continuous_on (W0 \<union> S) r"
       using continuous_on_subset contr sup_assoc by blast
-  qed auto
+  qed (use \<open>r ` (W0 \<union> S) \<subseteq> S\<close> in auto)
   have cloT'WT: "closedin (top_of_set T') (W0 \<union> T)"
     by (meson closedin_subset_trans UT cloUT' \<open>T \<subseteq> T'\<close> \<open>W \<subseteq> T'\<close> cloUW cloWW0 
               closedin_Un closedin_imp_subset closedin_trans)
   obtain W2 h where "W0 \<union> T \<subseteq> W2" and conth: "continuous_on W2 h"
                 and opeSW2: "openin (top_of_set T') W2"
                 and "h ` W2 \<subseteq> T" and heqr: "\<And>x. x \<in> W0 \<union> T \<Longrightarrow> h x = r x"
-  proof (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> _ \<open>r ` (W0 \<union> T) \<subseteq> T\<close> cloT'WT])
+  proof (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> _ _ cloT'WT])
     show "continuous_on (W0 \<union> T) r"
       using continuous_on_subset contr sup_assoc by blast
-  qed auto
+  qed (use \<open>r ` (W0 \<union> T) \<subseteq> T\<close> in auto)
   have "S' \<inter> T' = W"
     by (force simp: S'_def T'_def W_def)
   obtain O1 O2 where O12: "open O1" "W1 = S' \<inter> O1" "open O2" "W2 = T' \<inter> O2"
@@ -961,12 +962,12 @@
         using continuous_on_cases_local [OF cloW1 cloW2 continuous_on_subset [OF contg] continuous_on_subset [OF conth]]
         by simp
       show "?gh ` (?WW1 \<union> ?WW2) \<subseteq> S \<union> T"
-        using \<open>W1 = S' \<inter> O1\<close> \<open>W2 = T' \<inter> O2\<close> \<open>S' \<inter> T' = W\<close> \<open>g ` W1 \<subseteq> S\<close> \<open>h ` W2 \<subseteq> T\<close> \<open>U0 \<inter> W \<subseteq> W0\<close> \<open>W0 \<union> S \<subseteq> W1\<close>  
+        using \<open>W1 = S' \<inter> O1\<close> \<open>W2 = T' \<inter> O2\<close> \<open>S' \<inter> T' = W\<close> \<open>g \<in> W1 \<rightarrow> S\<close> \<open>h ` W2 \<subseteq> T\<close> \<open>U0 \<inter> W \<subseteq> W0\<close> \<open>W0 \<union> S \<subseteq> W1\<close>  
         by (auto simp add: image_subset_iff)
     qed
     then show "S \<union> T retract_of ?WW1 \<union> ?WW2"
       using  \<open>W0 \<union> S \<subseteq> W1\<close> \<open>W0 \<union> T \<subseteq> W2\<close> ST opeUU0 U0
-      by (auto simp: retract_of_def retraction_def)
+      by (auto simp: retract_of_def retraction_def image_subset_iff_funcset)
   qed
 qed
 
@@ -1025,17 +1026,17 @@
   shows "ANR S"
 proof (clarsimp simp only: ANR_eq_absolute_neighbourhood_extensor)
   fix f :: "'a \<times> real \<Rightarrow> 'a" and U C
-  assume contf: "continuous_on C f" and fim: "f ` C \<subseteq> S"
+  assume contf: "continuous_on C f" and fim: "f \<in> C \<rightarrow> S"
      and cloUC: "closedin (top_of_set U) C"
-  have "f ` C \<subseteq> T"
+  have "f \<in> C \<rightarrow> T"
     using fim opeTS openin_imp_subset by blast
   obtain W g where "C \<subseteq> W"
                and UW: "openin (top_of_set U) W"
                and contg: "continuous_on W g"
-               and gim: "g ` W \<subseteq> T"
+               and gim: "g \<in> W \<rightarrow> T"
                and geq: "\<And>x. x \<in> C \<Longrightarrow> g x = f x"
-    using ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> contf \<open>f ` C \<subseteq> T\<close> cloUC] fim by auto
-  show "\<exists>V g. C \<subseteq> V \<and> openin (top_of_set U) V \<and> continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x\<in>C. g x = f x)"
+    using ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> contf \<open>f \<in> C \<rightarrow> T\<close> cloUC] fim by auto
+  show "\<exists>V g. C \<subseteq> V \<and> openin (top_of_set U) V \<and> continuous_on V g \<and> g \<in> V \<rightarrow> S \<and> (\<forall>x\<in>C. g x = f x)"
   proof (intro exI conjI)
     show "C \<subseteq> W \<inter> g -` S"
       using \<open>C \<subseteq> W\<close> fim geq by blast
@@ -1043,7 +1044,7 @@
       by (metis (mono_tags, lifting) UW contg continuous_openin_preimage gim opeTS openin_trans)
     show "continuous_on (W \<inter> g -` S) g"
       by (blast intro: continuous_on_subset [OF contg])
-    show "g ` (W \<inter> g -` S) \<subseteq> S"
+    show "g \<in> (W \<inter> g -` S) \<rightarrow> S"
       using gim by blast
     show "\<forall>x\<in>C. g x = f x"
       using geq by blast
@@ -1217,7 +1218,7 @@
   assumes "ANR S" "ANR T" shows "ANR(S \<times> T)"
 proof (clarsimp simp only: ANR_eq_absolute_neighbourhood_extensor)
   fix f :: " ('a \<times> 'b) \<times> real \<Rightarrow> 'a \<times> 'b" and U C
-  assume "continuous_on C f" and fim: "f ` C \<subseteq> S \<times> T"
+  assume "continuous_on C f" and fim: "f \<in> C \<rightarrow> S \<times> T"
      and cloUC: "closedin (top_of_set U) C"
   have contf1: "continuous_on C (fst \<circ> f)"
     by (simp add: \<open>continuous_on C f\<close> continuous_on_fst)
@@ -1227,23 +1228,23 @@
                and gim: "g ` W1 \<subseteq> S"
                and geq: "\<And>x. x \<in> C \<Longrightarrow> g x = (fst \<circ> f) x"
   proof (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> contf1 _ cloUC])
-    show "(fst \<circ> f) ` C \<subseteq> S"
-      using fim by auto
+    show "(fst \<circ> f) \<in> C \<rightarrow> S"
+      using fim by force
   qed auto
   have contf2: "continuous_on C (snd \<circ> f)"
     by (simp add: \<open>continuous_on C f\<close> continuous_on_snd)
   obtain W2 h where "C \<subseteq> W2"
                and UW2: "openin (top_of_set U) W2"
                and conth: "continuous_on W2 h"
-               and him: "h ` W2 \<subseteq> T"
+               and him: "h \<in> W2 \<rightarrow> T"
                and heq: "\<And>x. x \<in> C \<Longrightarrow> h x = (snd \<circ> f) x"
   proof (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> contf2 _ cloUC])
-    show "(snd \<circ> f) ` C \<subseteq> T"
-      using fim by auto
+    show "(snd \<circ> f) \<in> C \<rightarrow> T"
+      using fim by force
   qed auto
   show "\<exists>V g. C \<subseteq> V \<and>
                openin (top_of_set U) V \<and>
-               continuous_on V g \<and> g ` V \<subseteq> S \<times> T \<and> (\<forall>x\<in>C. g x = f x)"
+               continuous_on V g \<and> g \<in> V \<rightarrow> S \<times> T \<and> (\<forall>x\<in>C. g x = f x)"
   proof (intro exI conjI)
     show "C \<subseteq> W1 \<inter> W2"
       by (simp add: \<open>C \<subseteq> W1\<close> \<open>C \<subseteq> W2\<close>)
@@ -1251,7 +1252,7 @@
       by (simp add: UW1 UW2 openin_Int)
     show  "continuous_on (W1 \<inter> W2) (\<lambda>x. (g x, h x))"
       by (metis (no_types) contg conth continuous_on_Pair continuous_on_subset inf_commute inf_le1)
-    show  "(\<lambda>x. (g x, h x)) ` (W1 \<inter> W2) \<subseteq> S \<times> T"
+    show  "(\<lambda>x. (g x, h x)) \<in> (W1 \<inter> W2) \<rightarrow> S \<times> T"
       using gim him by blast
     show  "(\<forall>x\<in>C. (g x, h x) = f x)"
       using geq heq by auto
@@ -1710,7 +1711,7 @@
   assumes cloTS: "closedin (top_of_set T) S"
       and anr: "(ANR S \<and> ANR T) \<or> ANR U"
       and contf: "continuous_on T f"
-      and "f ` T \<subseteq> U"
+      and "f \<in> T \<rightarrow> U"
       and "homotopic_with_canon (\<lambda>x. True) S U f g"
    obtains g' where "homotopic_with_canon (\<lambda>x. True) T U f g'"
                     "continuous_on T g'" "image g' T \<subseteq> U"
@@ -1718,9 +1719,9 @@
 proof -
   have "S \<subseteq> T" using assms closedin_imp_subset by blast
   obtain h where conth: "continuous_on ({0..1} \<times> S) h"
-             and him: "h ` ({0..1} \<times> S) \<subseteq> U"
+             and him: "h \<in> ({0..1} \<times> S) \<rightarrow> U"
              and [simp]: "\<And>x. h(0, x) = f x" "\<And>x. h(1::real, x) = g x"
-       using assms by (auto simp: homotopic_with_def)
+       using assms by (fastforce simp: homotopic_with_def)
   define h' where "h' \<equiv>  \<lambda>z. if snd z \<in> S then h z else (f \<circ> snd) z"
   define B where "B \<equiv> {0::real} \<times> T \<union> {0..1} \<times> S"
   have clo0T: "closedin (top_of_set ({0..1} \<times> T)) ({0::real} \<times> T)"
@@ -1744,9 +1745,9 @@
   then have conth': "continuous_on B h'"
     by (simp add: h'_def B_def Un_commute [of "{0} \<times> T"])
   have "image h' B \<subseteq> U"
-    using \<open>f ` T \<subseteq> U\<close> him by (auto simp: h'_def B_def)
+    using \<open>f \<in> T \<rightarrow> U\<close> him by (auto simp: h'_def B_def)
   obtain V k where "B \<subseteq> V" and opeTV: "openin (top_of_set ({0..1} \<times> T)) V"
-               and contk: "continuous_on V k" and kim: "k ` V \<subseteq> U"
+               and contk: "continuous_on V k" and kim: "k \<in> V \<rightarrow> U"
                and keq: "\<And>x. x \<in> B \<Longrightarrow> k x = h' x"
   using anr
   proof
@@ -1770,14 +1771,15 @@
       moreover have "(h' \<circ> r) ` V \<subseteq> U"
         by (metis \<open>h' ` B \<subseteq> U\<close> image_comp retractionE that(2))
       ultimately show ?thesis
-        using Vk [of V "h' \<circ> r"] by (metis comp_apply retraction that)
+        using Vk [of V "h' \<circ> r"] by (metis comp_apply retraction image_subset_iff_funcset that)
     qed
     show thesis
       by (meson "*" ANR_imp_neighbourhood_retract \<open>ANR B\<close> clo0TB retract_of_def)
   next
     assume "ANR U"
-    with ANR_imp_absolute_neighbourhood_extensor \<open>h' ` B \<subseteq> U\<close> clo0TB conth' that
-    show ?thesis by blast
+    with ANR_imp_absolute_neighbourhood_extensor \<open>h' ` B \<subseteq> U\<close> clo0TB conth' image_subset_iff_funcset that
+    show ?thesis
+      by (smt (verit) Pi_I funcset_mem) 
   qed
   define S' where "S' \<equiv> {x. \<exists>u::real. u \<in> {0..1} \<and> (u, x::'a) \<in> {0..1} \<times> T - V}"
   have "closedin (top_of_set T) S'"
@@ -1827,7 +1829,7 @@
     fix t
     assume "t \<in> T"
     show "k (a t, t) \<in> U"
-      by (metis \<open>t \<in> T\<close> image_subset_iff inV kim not_one_le_zero linear mult_cancel_right1)
+      by (metis \<open>t \<in> T\<close> image_subset_iff inV kim not_one_le_zero linear mult_cancel_right1 image_subset_iff_funcset)
   qed
   show "\<And>x. x \<in> S \<Longrightarrow> k (a x, x) = g x"
     by (simp add: B_def a1 h'_def keq)
@@ -1854,7 +1856,7 @@
   then obtain g where "continuous_on UNIV g" "range g \<subseteq> T"
                       "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
   proof (rule Borsuk_homotopy_extension_homotopic)
-    show "range (\<lambda>x. c) \<subseteq> T"
+    show "(\<lambda>x. c) \<in> UNIV \<rightarrow> T"
       using \<open>S \<noteq> {}\<close> c homotopic_with_imp_subset1 by fastforce
   qed (use assms c in auto)
   then show ?rhs by blast
@@ -1956,7 +1958,7 @@
         by (metis convex_imp_contractible convex_ball)
       show "continuous_on (ball 0 r) (\<lambda>x. inverse(norm (x - b)) *\<^sub>R (x - b))"
         by (rule continuous_on_Borsuk_map [OF bnot])
-      show "(\<lambda>x. (x - b) /\<^sub>R norm (x - b)) ` ball 0 r \<subseteq> sphere 0 1"
+      show "(\<lambda>x. (x - b) /\<^sub>R norm (x - b)) \<in> ball 0 r \<rightarrow> sphere 0 1"
         using bnot Borsuk_map_into_sphere by blast
     qed blast
     ultimately have "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) (\<lambda>x. (x - a) /\<^sub>R norm (x - a)) (\<lambda>x. c)"
@@ -1979,9 +1981,10 @@
   assume notcc: "\<not> connected_component (- S) a b"
   let ?T = "S \<union> connected_component_set (- S) a"
   have "\<nexists>g. continuous_on (S \<union> connected_component_set (- S) a) g \<and>
-            g ` (S \<union> connected_component_set (- S) a) \<subseteq> sphere 0 1 \<and>
+            g \<in> (S \<union> connected_component_set (- S) a) \<rightarrow> sphere 0 1 \<and>
             (\<forall>x\<in>S. g x = (x - a) /\<^sub>R norm (x - a))"
-    by (simp add: \<open>a \<notin> S\<close> componentsI non_extensible_Borsuk_map [OF \<open>compact S\<close> _ boc])
+    using non_extensible_Borsuk_map [OF \<open>compact S\<close> _ boc] \<open>a \<notin> S\<close>
+    by (simp add: componentsI)
   moreover obtain g where "continuous_on (S \<union> connected_component_set (- S) a) g"
                           "g ` (S \<union> connected_component_set (- S) a) \<subseteq> sphere 0 1"
                           "\<And>x. x \<in> S \<Longrightarrow> g x = (x - a) /\<^sub>R norm (x - a)"
@@ -1990,7 +1993,7 @@
       by (simp add: \<open>compact S\<close> closed_subset compact_imp_closed)
     show "continuous_on ?T (\<lambda>x. (x - b) /\<^sub>R norm (x - b))"
       by (simp add: \<open>b \<notin> S\<close> notcc continuous_on_Borsuk_map)
-    show "(\<lambda>x. (x - b) /\<^sub>R norm (x - b)) ` ?T \<subseteq> sphere 0 1"
+    show "(\<lambda>x. (x - b) /\<^sub>R norm (x - b)) \<in> ?T \<rightarrow> sphere 0 1"
       by (simp add: \<open>b \<notin> S\<close> notcc Borsuk_map_into_sphere)
     show "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1)
              (\<lambda>x. (x - b) /\<^sub>R norm (x - b)) (\<lambda>x. (x - a) /\<^sub>R norm (x - a))"
@@ -2073,12 +2076,12 @@
 lemma extension_from_component:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   assumes S: "locally connected S \<or> compact S" and "ANR U"
-     and C: "C \<in> components S" and contf: "continuous_on C f" and fim: "f ` C \<subseteq> U"
- obtains g where "continuous_on S g" "g ` S \<subseteq> U" "\<And>x. x \<in> C \<Longrightarrow> g x = f x"
+     and C: "C \<in> components S" and contf: "continuous_on C f" and fim: "f \<in> C \<rightarrow> U"
+ obtains g where "continuous_on S g" "g \<in> S \<rightarrow> U" "\<And>x. x \<in> C \<Longrightarrow> g x = f x"
 proof -
   obtain T g where ope: "openin (top_of_set S) T"
                and clo: "closedin (top_of_set S) T"
-               and "C \<subseteq> T" and contg: "continuous_on T g" and gim: "g ` T \<subseteq> U"
+               and "C \<subseteq> T" and contg: "continuous_on T g" and gim: "g \<in> T \<rightarrow> U"
                and gf: "\<And>x. x \<in> C \<Longrightarrow> g x = f x"
     using S
   proof
@@ -2089,7 +2092,7 @@
     assume "compact S"
     then obtain W g where "C \<subseteq> W" and opeW: "openin (top_of_set S) W"
                  and contg: "continuous_on W g"
-                 and gim: "g ` W \<subseteq> U" and gf: "\<And>x. x \<in> C \<Longrightarrow> g x = f x"
+                 and gim: "g \<in> W \<rightarrow> U" and gf: "\<And>x. x \<in> C \<Longrightarrow> g x = f x"
       using ANR_imp_absolute_neighbourhood_extensor [of U C f S] C \<open>ANR U\<close> closedin_component contf fim by blast
     then obtain V where "open V" and V: "W = S \<inter> V"
       by (auto simp: openin_open)
@@ -2103,13 +2106,13 @@
         by (meson \<open>compact K\<close> \<open>compact S\<close> closedin_compact_eq opeK openin_imp_subset)
       show "continuous_on K g"
         by (metis Int_subset_iff V \<open>K \<subseteq> V\<close> contg continuous_on_subset opeK openin_subtopology subset_eq)
-      show "g ` K \<subseteq> U"
+      show "g \<in> K \<rightarrow> U"
         using V \<open>K \<subseteq> V\<close> gim opeK openin_imp_subset by fastforce
     qed (use opeK gf \<open>C \<subseteq> K\<close> in auto)
   qed
-  obtain h where "continuous_on S h" "h ` S \<subseteq> U" "\<And>x. x \<in> T \<Longrightarrow> h x = g x"
+  obtain h where "continuous_on S h" "h \<in> S \<rightarrow> U" "\<And>x. x \<in> T \<Longrightarrow> h x = g x"
     using extension_from_clopen
-    by (metis C bot.extremum_uniqueI clo contg gim fim image_is_empty in_components_nonempty ope)
+    by (metis C bot.extremum_uniqueI clo contg gim fim image_is_empty in_components_nonempty ope image_subset_iff_funcset)
   then show ?thesis
     by (metis \<open>C \<subseteq> T\<close> gf subset_eq that)
 qed
@@ -2198,9 +2201,9 @@
   obtain W k where W: "({0,1} \<times> S) \<union> ({0..1} \<times> T) \<subseteq> W"
                and opeW: "openin (top_of_set ({0..1} \<times> S)) W"
                and contk: "continuous_on W k"
-               and kim: "k ` W \<subseteq> U"
+               and kim: "k \<in> W \<rightarrow> U"
                and kh': "\<And>x. x \<in> ({0,1} \<times> S) \<union> ({0..1} \<times> T) \<Longrightarrow> k x = h' x"
-    by (metis ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR U\<close>, of "({0,1} \<times> S) \<union> ({0..1} \<times> T)" h' "{0..1} \<times> S"])
+    by (metis ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR U\<close>, of "({0,1} \<times> S) \<union> ({0..1} \<times> T)" h' "{0..1} \<times> S"] image_subset_iff_funcset)
   obtain T' where opeT': "openin (top_of_set S) T'" 
               and "T \<subseteq> T'" and TW: "{0..1} \<times> T' \<subseteq> W"
     using tube_lemma_gen [of "{0..1::real}" T S W] W \<open>T \<subseteq> S\<close> opeW by auto
@@ -2377,11 +2380,11 @@
   fixes S :: "'a :: euclidean_space set" and T :: "'b :: euclidean_space set"
   assumes S: "locally connected S \<or> compact S" and "ANR T"
   shows
-   "(\<forall>f g. continuous_on S f \<longrightarrow> f ` S \<subseteq> T \<longrightarrow> continuous_on S g \<longrightarrow> g ` S \<subseteq> T \<longrightarrow>
+   "(\<forall>f g. continuous_on S f \<longrightarrow> f \<in> S \<rightarrow> T \<longrightarrow> continuous_on S g \<longrightarrow> g \<in> S \<rightarrow> T \<longrightarrow>
            homotopic_with_canon (\<lambda>x. True) S T f g)
     \<longleftrightarrow>
     (\<forall>C\<in>components S.
-        \<forall>f g. continuous_on C f \<longrightarrow> f ` C \<subseteq> T \<longrightarrow> continuous_on C g \<longrightarrow> g ` C \<subseteq> T \<longrightarrow>
+        \<forall>f g. continuous_on C f \<longrightarrow> f \<in> C \<rightarrow> T \<longrightarrow> continuous_on C g \<longrightarrow> g \<in> C \<rightarrow> T \<longrightarrow>
               homotopic_with_canon (\<lambda>x. True) C T f g)"
      (is "?lhs = ?rhs")
 proof
@@ -2389,11 +2392,11 @@
   show ?rhs
   proof clarify
     fix C f g
-    assume contf: "continuous_on C f" and fim: "f ` C \<subseteq> T"
-       and contg: "continuous_on C g" and gim: "g ` C \<subseteq> T" and C: "C \<in> components S"
-    obtain f' where contf': "continuous_on S f'" and f'im: "f' ` S \<subseteq> T" and f'f: "\<And>x. x \<in> C \<Longrightarrow> f' x = f x"
+    assume contf: "continuous_on C f" and fim: "f \<in> C \<rightarrow> T"
+       and contg: "continuous_on C g" and gim: "g \<in> C \<rightarrow> T" and C: "C \<in> components S"
+    obtain f' where contf': "continuous_on S f'" and f'im: "f' \<in> S \<rightarrow> T" and f'f: "\<And>x. x \<in> C \<Longrightarrow> f' x = f x"
       using extension_from_component [OF S \<open>ANR T\<close> C contf fim] by metis
-    obtain g' where contg': "continuous_on S g'" and g'im: "g' ` S \<subseteq> T" and g'g: "\<And>x. x \<in> C \<Longrightarrow> g' x = g x"
+    obtain g' where contg': "continuous_on S g'" and g'im: "g' \<in> S \<rightarrow> T" and g'g: "\<And>x. x \<in> C \<Longrightarrow> g' x = g x"
       using extension_from_component [OF S \<open>ANR T\<close> C contg gim] by metis
     have "homotopic_with_canon (\<lambda>x. True) C T f' g'"
       using L [OF contf' f'im contg' g'im] homotopic_with_subset_left C in_components_subset by fastforce
@@ -2405,11 +2408,11 @@
   show ?lhs
   proof clarify
     fix f g
-    assume contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
-      and contg: "continuous_on S g" and gim: "g ` S \<subseteq> T"
+    assume contf: "continuous_on S f" and fim: "f \<in> S \<rightarrow> T"
+      and contg: "continuous_on S g" and gim: "g \<in> S \<rightarrow> T"
     moreover have "homotopic_with_canon (\<lambda>x. True) C T f g" if "C \<in> components S" for C
-      using R [OF that]
-      by (meson contf contg continuous_on_subset fim gim image_mono in_components_subset order.trans that)
+      using R [OF that] contf contg continuous_on_subset fim gim in_components_subset
+      by (smt (verit, del_insts) Pi_anti_mono subsetD that)
     ultimately show "homotopic_with_canon (\<lambda>x. True) S T f g"
       by (subst homotopic_on_components_eq [OF S \<open>ANR T\<close>]) auto
   qed
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Starlike.thy
--- a/src/HOL/Analysis/Starlike.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Starlike.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -4202,7 +4202,7 @@
 
 lemma proper_map_from_compact:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T" and "compact S"
+  assumes contf: "continuous_on S f" and imf: "f \<in> S \<rightarrow> T" and "compact S"
           "closedin (top_of_set T) K"
   shows "compact (S \<inter> f -` K)"
 by (rule closedin_compact [OF \<open>compact S\<close>] continuous_closedin_preimage_gen assms)+
@@ -5820,7 +5820,7 @@
 
 lemma continuous_closed_graph_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "compact T" and fim: "f ` S \<subseteq> T"
+  assumes "compact T" and fim: "f \<in> S \<rightarrow> T"
   shows "continuous_on S f \<longleftrightarrow>
          closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
          (is "?lhs = ?rhs")
@@ -5848,7 +5848,7 @@
 
 lemma continuous_from_closed_graph:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "compact T" and fim: "f ` S \<subseteq> T" and clo: "closed ((\<lambda>x. Pair x (f x)) ` S)"
+  assumes "compact T" and fim: "f \<in> S \<rightarrow> T" and clo: "closed ((\<lambda>x. Pair x (f x)) ` S)"
   shows "continuous_on S f"
     using fim clo
     by (auto intro: closed_subset simp: continuous_closed_graph_eq [OF \<open>compact T\<close> fim])
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Analysis/Urysohn.thy
--- a/src/HOL/Analysis/Urysohn.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Analysis/Urysohn.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -2516,8 +2516,9 @@
   show ?thesis
     unfolding Lipschitz_continuous_map_pos
   proof
-    show "f ` mspace (submetric m1 T) \<subseteq> mspace m2"
-      by (metis cmf Metric_space.metric_continuous_map Metric_space_mspace_mdist  mtopology_of_def mtopology_of_submetric)
+    show "f \<in> mspace (submetric m1 T) \<rightarrow> mspace m2"
+      by (metis cmf Metric_space.metric_continuous_map Metric_space_mspace_mdist  mtopology_of_def 
+          mtopology_of_submetric image_subset_iff_funcset)
     define X where "X \<equiv> prod_topology (subtopology L.M1.mtopology T) (subtopology L.M1.mtopology T)"
     obtain B::real where "B > 0" and B: "\<forall>(x,y) \<in> S\<times>S. mdist m2 (f x) (f y) \<le> B * mdist m1 x y"
       using lcf \<open>S \<subseteq> mspace m1\<close>  by (force simp: Lipschitz_continuous_map_pos)
@@ -2608,8 +2609,9 @@
   show ?thesis
     unfolding uniformly_continuous_map_def
     proof (intro conjI strip)
-    show "f ` mspace (submetric m1 T) \<subseteq> mspace m2"
-      by (metis cmf Metric_space.metric_continuous_map Metric_space_mspace_mdist  mtopology_of_def mtopology_of_submetric)
+    show "f \<in> mspace (submetric m1 T) \<rightarrow> mspace m2"
+      by (metis cmf Metric_space.metric_continuous_map Metric_space_mspace_mdist  
+          mtopology_of_def mtopology_of_submetric image_subset_iff_funcset)
     fix \<epsilon>::real
     assume "\<epsilon> > 0"
     then obtain \<delta> where "\<delta>>0" and \<delta>: "\<forall>(x,y) \<in> S\<times>S. mdist m1 x y < \<delta> \<longrightarrow> mdist m2 (f x) (f y) \<le> \<epsilon>/2"
@@ -3143,7 +3145,7 @@
 
 lemma mspace_cfunspace [simp]:
   "mspace (cfunspace X m) = 
-     {f. f ` topspace X \<subseteq> mspace m \<and> f \<in> extensional (topspace X) \<and>
+     {f. f \<in> topspace X \<rightarrow> mspace m \<and> f \<in> extensional (topspace X) \<and>
          Metric_space.mbounded (mspace m) (mdist m) (f ` (topspace X)) \<and>
          continuous_map X (mtopology_of m) f}"
   by (auto simp: cfunspace_def Metric_space.fspace_def)
@@ -3278,7 +3280,7 @@
       assume g: "range \<sigma> \<subseteq> mspace (cfunspace X Self) \<and> limitin F.mtopology \<sigma> g sequentially"
       show "g \<in> mspace (cfunspace X Self)"
       proof (simp add: mtopology_of_def, intro conjI)
-        show "g ` topspace X \<subseteq> M" "g \<in> extensional (topspace X)" "mbounded (g ` topspace X)"
+        show "g \<in> topspace X \<rightarrow> M" "g \<in> extensional (topspace X)" "mbounded (g ` topspace X)"
           using g F.limitin_mspace by (force simp: fspace_def)+
         show "continuous_map X mtopology g"
           using * g by blast
@@ -3296,7 +3298,7 @@
   obtains f :: "['a,'a] \<Rightarrow> real" and S where
       "S \<subseteq> mspace(funspace M euclidean_metric)"
       "mcomplete_of (submetric (funspace M euclidean_metric) S)"
-      "f ` M \<subseteq> S"
+      "f \<in> M \<rightarrow> S"
       "mtopology_of(funspace M euclidean_metric) closure_of f ` M = S"
       "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk>
             \<Longrightarrow> mdist (funspace M euclidean_metric) (f x) (f y) = d x y"
@@ -3337,7 +3339,7 @@
         using Metric_space.mcomplete_funspace Met_TC.Metric_space_axioms by (fastforce simp: mcomplete_of_def)
       ultimately show "mcomplete_of (submetric (funspace M euclidean_metric) S)"
         by (simp add: S.closedin_eq_mcomplete mcomplete_of_def)
-      show "f ` M \<subseteq> S"
+      show "f \<in> M \<rightarrow> S"
         using S_def fim in_closure_of m'_def by fastforce
       show "mtopology_of (funspace M euclidean_metric) closure_of f ` M = S"
         by (auto simp: f_def S_def mtopology_of_def)
@@ -3363,14 +3365,14 @@
 lemma (in Metric_space) metric_completion:
   obtains f :: "['a,'a] \<Rightarrow> real" and m' where
     "mcomplete_of m'"
-    "f ` M \<subseteq> mspace m' "
+    "f \<in> M \<rightarrow> mspace m' "
     "mtopology_of m' closure_of f ` M = mspace m'"
     "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> mdist m' (f x) (f y) = d x y"
 proof -
   obtain f :: "['a,'a] \<Rightarrow> real" and S where
     Ssub: "S \<subseteq> mspace(funspace M euclidean_metric)"
     and mcom: "mcomplete_of (submetric (funspace M euclidean_metric) S)"
-    and fim: "f ` M \<subseteq> S"
+    and fim: "f \<in> M \<rightarrow> S"
     and eqS: "mtopology_of(funspace M euclidean_metric) closure_of f ` M = S"
     and eqd: "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> mdist (funspace M euclidean_metric) (f x) (f y) = d x y"
     using metric_completion_explicit by metis
@@ -3379,11 +3381,11 @@
   proof
     show "mcomplete_of m'"
       by (simp add: mcom m'_def)
-    show "f ` M \<subseteq> mspace m'"
+    show "f \<in> M \<rightarrow> mspace m'"
       using Ssub fim m'_def by auto
     show "mtopology_of m' closure_of f ` M = mspace m'"
       using eqS fim Ssub
-      by (force simp: m'_def mtopology_of_submetric closure_of_subtopology Int_absorb1)
+      by (force simp: m'_def mtopology_of_submetric closure_of_subtopology Int_absorb1 image_subset_iff_funcset)
     show "mdist m' (f x) (f y) = d x y" if "x \<in> M" and "y \<in> M" for x y
       using that eqd m'_def by force 
   qed 
@@ -3400,10 +3402,10 @@
   then interpret Metric_space M d by simp
   obtain f :: "['a,'a] \<Rightarrow> real" and m' where
     "mcomplete_of m'"
-    and fim: "f ` M \<subseteq> mspace m' "
+    and fim: "f \<in> M \<rightarrow> mspace m' "
     and m': "mtopology_of m' closure_of f ` M = mspace m'"
     and eqd: "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> mdist m' (f x) (f y) = d x y"
-    by (meson metric_completion)
+    by (metis metric_completion)
   show thesis
   proof
     show "completely_metrizable_space (mtopology_of m')"
@@ -3412,7 +3414,8 @@
       by (metis Metric_space_mspace_mdist)
     show "embedding_map X (mtopology_of m') f"
       using Metric_space12.isometry_imp_embedding_map
-      by (metis Metric_space12_def Metric_space_axioms Metric_space_mspace_mdist Xeq eqd fim mtopology_of_def)
+      by (metis Metric_space12_def Metric_space_axioms Metric_space_mspace_mdist Xeq eqd fim 
+             mtopology_of_def)
     show "(mtopology_of m') closure_of f ` topspace X = topspace (mtopology_of m')"
       by (simp add: Xeq m')
   qed
@@ -3424,7 +3427,7 @@
 
 lemma (in Metric_space) contraction_imp_unique_fixpoint:
   assumes "f x = x" "f y = y"
-    and "f ` M \<subseteq> M"
+    and "f \<in> M \<rightarrow> M"
     and "k < 1"
     and "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> d (f x) (f y) \<le> k * d x y"
     and "x \<in> M" "y \<in> M"
@@ -3433,7 +3436,7 @@
 
 text \<open>Banach Fixed-Point Theorem (aka, Contraction Mapping Principle)\<close>
 lemma (in Metric_space) Banach_fixedpoint_thm:
-  assumes mcomplete and "M \<noteq> {}" and fim: "f ` M \<subseteq> M"    
+  assumes mcomplete and "M \<noteq> {}" and fim: "f \<in> M \<rightarrow> M"    
     and "k < 1"
     and con: "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> d (f x) (f y) \<le> k * d x y"
   obtains x where "x \<in> M" "f x = x"
@@ -3444,7 +3447,7 @@
   proof (cases "\<forall>x \<in> M. f x = f a")
     case True
     then show ?thesis
-      by (metis \<open>a \<in> M\<close> fim image_subset_iff that)
+      by (metis \<open>a \<in> M\<close> fim image_subset_iff image_subset_iff_funcset that)
   next
     case False
     then obtain b where "b \<in> M" and b: "f b \<noteq> f a"
@@ -3472,7 +3475,7 @@
           fix \<epsilon>::real
           assume "\<epsilon>>0"
           with \<open>k < 1\<close> \<open>f a \<noteq> a\<close> \<open>a \<in> M\<close> fim have gt0: "((1 - k) * \<epsilon>) / d a (f a) > 0"
-            by (fastforce simp: divide_simps)
+            by (fastforce simp: divide_simps Pi_iff)
           obtain N where "k^N < ((1-k) * \<epsilon>) / d a (f a)"
             using real_arch_pow_inv [OF gt0 \<open>k < 1\<close>] by blast
           then have N: "\<And>n. n \<ge> N \<Longrightarrow> k^n < ((1-k) * \<epsilon>) / d a (f a)"
@@ -3992,11 +3995,11 @@
   shows "embedding_map X
              (product_topology (\<lambda>f. top_of_set {0..1::real})
                 (mspace (submetric (cfunspace X euclidean_metric)
-                  {f. f ` topspace X \<subseteq> {0..1}})) )
-             (\<lambda>x. \<lambda>f \<in> mspace (submetric (cfunspace X euclidean_metric) {f. f ` topspace X \<subseteq> {0..1}}).
+                  {f. f \<in> topspace X \<rightarrow> {0..1}})) )
+             (\<lambda>x. \<lambda>f \<in> mspace (submetric (cfunspace X euclidean_metric) {f. f \<in> topspace X \<rightarrow> {0..1}}).
               f x)"
 proof -
-  define K where "K \<equiv> mspace(submetric (cfunspace X euclidean_metric) {f. f ` topspace X \<subseteq> {0..1::real}})"
+  define K where "K \<equiv> mspace(submetric (cfunspace X euclidean_metric) {f. f \<in> topspace X \<rightarrow> {0..1::real}})"
   define e where "e \<equiv> \<lambda>x. \<lambda>f\<in>K. f x"
   have "e x \<noteq> e y" if xy: "x \<noteq> y" "x \<in> topspace X" "y \<in> topspace X" for x y
   proof -
@@ -4023,14 +4026,14 @@
     assume "e x \<in> K \<rightarrow>\<^sub>E {0..1}" and "x \<in> topspace X" and "openin X U" and "x \<in> U"
     then obtain g where contg: "continuous_map X (top_of_set {0..1}) g" and "g x = 0" 
           and gim: "g ` (topspace X - U) \<subseteq> {1::real}"
-      using \<open>completely_regular_space X\<close> unfolding completely_regular_space_def
+      using \<open>completely_regular_space X\<close> unfolding completely_regular_space_def 
       by (metis Diff_iff openin_closedin_eq)
     then have "bounded (g ` topspace X)"
       by (meson bounded_closed_interval bounded_subset continuous_map_in_subtopology)
-    moreover have "g ` topspace X \<subseteq> {0..1}"
-      using contg by (simp add: continuous_map_in_subtopology)
+    moreover have "g \<in> topspace X \<rightarrow> {0..1}"
+      using contg by (simp add: continuous_map_def)
     ultimately have g_in_K: "restrict g (topspace X) \<in> K"
-      using contg by (simp add: K_def continuous_map_in_subtopology)
+      using contg by (force simp add: K_def continuous_map_in_subtopology)
     have "openin (top_of_set {0..1}) {0..<1::real}"
       using open_real_greaterThanLessThan[of "-1" 1] by (force simp: openin_open)
     moreover have "e x \<in> (\<Pi>\<^sub>E f\<in>K. if f = restrict g (topspace X) then {0..<1} else {0..1})"
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Complex_Analysis/Conformal_Mappings.thy
--- a/src/HOL/Complex_Analysis/Conformal_Mappings.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Complex_Analysis/Conformal_Mappings.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -1145,7 +1145,7 @@
 lemma covering_space_lift_is_holomorphic:
   assumes cov: "covering_space C p S"
       and C: "open C" "p holomorphic_on C"
-      and holf: "f holomorphic_on U" and fim: "f ` U \<subseteq> S" and gim: "g ` U \<subseteq> C"
+      and holf: "f holomorphic_on U" and fim: "f \<in> U \<rightarrow> S" and gim: "g \<in> U \<rightarrow> C"
       and contg: "continuous_on U g" and pg_f: "\<And>x. x \<in> U \<Longrightarrow> p(g x) = f x"
     shows "g holomorphic_on U"
   unfolding holomorphic_on_def
@@ -1161,7 +1161,7 @@
   then have "\<And>W. W \<in> \<V> \<Longrightarrow> open W \<and> W \<subseteq> C"
     by (metis \<open>open C\<close> inf_le1 open_Int openin_open) 
   then obtain V where "V \<in> \<V>" "g z \<in> V" "open V" "V \<subseteq> C"
-    by (metis IntI UnionE image_subset_iff vimageI UV \<open>f z \<in> T\<close> \<open>z \<in> U\<close> gim pg_f)
+    by (metis IntI UnionE image_subset_iff vimageI UV \<open>f z \<in> T\<close> \<open>z \<in> U\<close> gim pg_f image_subset_iff_funcset)
   have holp: "p holomorphic_on V"
     using \<open>V \<subseteq> C\<close> \<open>p holomorphic_on C\<close> holomorphic_on_subset by blast
   moreover have injp: "inj_on p V"
@@ -1201,15 +1201,14 @@
 lemma covering_space_lift_holomorphic:
   assumes cov: "covering_space C p S"
       and C: "open C" "p holomorphic_on C"
-      and f: "f holomorphic_on U" "f ` U \<subseteq> S" 
+      and f: "f holomorphic_on U" "f \<in> U \<rightarrow> S" 
       and U: "simply_connected U" "locally path_connected U"
-    obtains g where  "g holomorphic_on U" "g ` U \<subseteq> C" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
+  obtains g where "g holomorphic_on U" "g \<in> U \<rightarrow> C" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
 proof -
-  obtain g where "continuous_on U g" "g ` U \<subseteq> C" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
-    using covering_space_lift [OF cov U]
-    using f holomorphic_on_imp_continuous_on by blast
+  obtain g where "continuous_on U g" "g \<in> U \<rightarrow> C" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
+    using covering_space_lift [OF cov U] f holomorphic_on_imp_continuous_on by blast
   then show ?thesis
-    by (metis C cov covering_space_lift_is_holomorphic f that)
+    by (metis C cov covering_space_lift_is_holomorphic f image_subset_iff_funcset that)
 qed
 
 subsection\<open>The Schwarz Lemma\<close>
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Complex_Analysis/Riemann_Mapping.thy
--- a/src/HOL/Complex_Analysis/Riemann_Mapping.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Complex_Analysis/Riemann_Mapping.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -1624,7 +1624,7 @@
     qed
     show "continuous_on UNIV (\<lambda>w. \<zeta> + exp w)"
       by (rule continuous_intros)+
-    show "range (\<lambda>w. \<zeta> + exp w) \<subseteq> -{\<zeta>}"
+    show "(\<lambda>w. \<zeta> + exp w) \<in> UNIV \<rightarrow> -{\<zeta>}"
       by auto
   qed
   then have "homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r) {0..1} (-{\<zeta>}) p (\<lambda>x. \<zeta> + 1)"
diff -r d4125fc10c0c -r 1260be3f33a4 src/HOL/Library/FuncSet.thy
--- a/src/HOL/Library/FuncSet.thy	Sun Jul 02 20:41:07 2023 +0200
+++ b/src/HOL/Library/FuncSet.thy	Mon Jul 03 16:46:37 2023 +0100
@@ -148,9 +148,10 @@
   by (auto simp: Pi_def)
 
 lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A"
-  apply auto
-  apply (metis PiE fun_upd_apply)
-  by force
+  using mk_disjoint_insert by fastforce
+
+lemma fst_Pi: "fst \<in> A \<times> B \<rightarrow> A" and snd_Pi: "snd \<in> A \<times> B \<rightarrow> B"
+  by auto
 
 
 subsection \<open>Composition With a Restricted Domain: \<^term>\<open>compose\<close>\<close>