New lemmas and a bit of tidying up.

--- a/src/HOL/Finite_Set.thy	Tue Feb 10 12:27:30 2015 +0100
+++ b/src/HOL/Finite_Set.thy	Tue Feb 10 16:08:11 2015 +0000
@@ -262,6 +262,10 @@
   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
   by (simp add: image_Collect [symmetric])
 
+lemma finite_image_set2:
+  "finite {x. P x} \<Longrightarrow> finite {y. Q y} \<Longrightarrow> finite {f x y | x y. P x \<and> Q y}"
+  by (rule finite_subset [where B = "\<Union>x \<in> {x. P x}. \<Union>y \<in> {y. Q y}. {f x y}"]) auto
+
 lemma finite_imageD:
   assumes "finite (f ` A)" and "inj_on f A"
   shows "finite A"
@@ -1618,6 +1622,9 @@
    by (force intro: card_mono simp: card_image [symmetric])
 qed
 
+lemma surj_card_le: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> card B \<le> card A"
+  by (blast intro: card_image_le card_mono le_trans)
+
 lemma card_bij_eq:
   "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
      finite A; finite B |] ==> card A = card B"

--- a/src/HOL/Fun.thy	Tue Feb 10 12:27:30 2015 +0100
+++ b/src/HOL/Fun.thy	Tue Feb 10 16:08:11 2015 +0000
@@ -15,6 +15,12 @@
   "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
   by auto
 
+text{*Uniqueness, so NOT the axiom of choice.*}
+lemma uniq_choice: "\<forall>x. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x. Q x (f x)"
+  by (force intro: theI')
+
+lemma b_uniq_choice: "\<forall>x\<in>S. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x\<in>S. Q x (f x)"
+  by (force intro: theI')
 
 subsection {* The Identity Function @{text id} *}
 
@@ -79,6 +85,9 @@
   "f -` (g -` x) = (g \<circ> f) -` x"
   by auto
 
+lemma image_eq_imp_comp: "f ` A = g ` B \<Longrightarrow> (h o f) ` A = (h o g) ` B"
+  by (auto simp: comp_def elim!: equalityE)
+
 code_printing
   constant comp \<rightharpoonup> (SML) infixl 5 "o" and (Haskell) infixr 9 "."
 
@@ -477,14 +486,17 @@
 lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
 by (simp add: inj_on_def, blast)
 
-lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
-by (blast dest: injD)
+lemma inj_on_image_mem_iff: "\<lbrakk>inj_on f B; a \<in> B; A \<subseteq> B\<rbrakk> \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A"
+  by (auto simp: inj_on_def)
+
+lemma inj_image_mem_iff: "inj f \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A"
+  by (blast dest: injD)
 
 lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
-by (simp add: inj_on_def, blast)
+  by (blast dest: injD)
 
 lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
-by (blast dest: injD)
+  by (blast dest: injD)
 
 lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
 by auto

--- a/src/HOL/HOL.thy	Tue Feb 10 12:27:30 2015 +0100
+++ b/src/HOL/HOL.thy	Tue Feb 10 16:08:11 2015 +0000
@@ -550,63 +550,6 @@
 done
 
 
-subsubsection {*THE: definite description operator*}
-
-lemma the_equality:
-  assumes prema: "P a"
-      and premx: "!!x. P x ==> x=a"
-  shows "(THE x. P x) = a"
-apply (rule trans [OF _ the_eq_trivial])
-apply (rule_tac f = "The" in arg_cong)
-apply (rule ext)
-apply (rule iffI)
- apply (erule premx)
-apply (erule ssubst, rule prema)
-done
-
-lemma theI:
-  assumes "P a" and "!!x. P x ==> x=a"
-  shows "P (THE x. P x)"
-by (iprover intro: assms the_equality [THEN ssubst])
-
-lemma theI': "EX! x. P x ==> P (THE x. P x)"
-apply (erule ex1E)
-apply (erule theI)
-apply (erule allE)
-apply (erule mp)
-apply assumption
-done
-
-(*Easier to apply than theI: only one occurrence of P*)
-lemma theI2:
-  assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
-  shows "Q (THE x. P x)"
-by (iprover intro: assms theI)
-
-lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
-by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
-           elim:allE impE)
-
-lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
-apply (rule the_equality)
-apply  assumption
-apply (erule ex1E)
-apply (erule all_dupE)
-apply (drule mp)
-apply  assumption
-apply (erule ssubst)
-apply (erule allE)
-apply (erule mp)
-apply assumption
-done
-
-lemma the_sym_eq_trivial: "(THE y. x=y) = x"
-apply (rule the_equality)
-apply (rule refl)
-apply (erule sym)
-done
-
-
 subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
 
 lemma disjCI:
@@ -876,7 +819,6 @@
 
 declare ex_ex1I [intro!]
   and allI [intro!]
-  and the_equality [intro]
   and exI [intro]
 
 declare exE [elim!]
@@ -924,6 +866,39 @@
 *}
 
 
+subsubsection {*THE: definite description operator*}
+
+lemma the_equality [intro]:
+  assumes "P a"
+      and "!!x. P x ==> x=a"
+  shows "(THE x. P x) = a"
+  by (blast intro: assms trans [OF arg_cong [where f=The] the_eq_trivial])
+
+lemma theI:
+  assumes "P a" and "!!x. P x ==> x=a"
+  shows "P (THE x. P x)"
+by (iprover intro: assms the_equality [THEN ssubst])
+
+lemma theI': "EX! x. P x ==> P (THE x. P x)"
+  by (blast intro: theI)
+
+(*Easier to apply than theI: only one occurrence of P*)
+lemma theI2:
+  assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
+  shows "Q (THE x. P x)"
+by (iprover intro: assms theI)
+
+lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
+by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
+           elim:allE impE)
+
+lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
+  by blast
+
+lemma the_sym_eq_trivial: "(THE y. x=y) = x"
+  by blast
+
+
 subsubsection {* Simplifier *}
 
 lemma eta_contract_eq: "(%s. f s) = f" ..
@@ -1099,8 +1074,7 @@
 
 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
   -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
-  apply (simplesubst split_if, blast)
-  done
+  by (simplesubst split_if) blast
 
 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover

--- a/src/HOL/Set.thy	Tue Feb 10 12:27:30 2015 +0100
+++ b/src/HOL/Set.thy	Tue Feb 10 16:08:11 2015 +0000
@@ -1020,6 +1020,9 @@
   "f ` A - f ` B \<subseteq> f ` (A - B)"
   by blast
 
+lemma Setcompr_eq_image: "{f x | x. x \<in> A} = f ` A"
+  by blast
+
 lemma ball_imageD:
   assumes "\<forall>x\<in>f ` A. P x"
   shows "\<forall>x\<in>A. P (f x)"
@@ -1241,6 +1244,9 @@
 lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
   by blast
 
+lemma Collect_mono_iff [simp]: "Collect P \<subseteq> Collect Q \<longleftrightarrow> (\<forall>x. P x \<longrightarrow> Q x)"
+  by blast
+
 
 text {* \medskip @{text insert}. *}