src/ZF/Perm.thy
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(*  Title: 	ZF/perm
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    ID:         $Id$
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    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1991  University of Cambridge
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The theory underlying permutation groups
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  -- Composition of relations, the identity relation
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  -- Injections, surjections, bijections
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  -- Lemmas for the Schroeder-Bernstein Theorem
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*)
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Perm = ZF + "mono" +
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consts
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    O    	::      "[i,i]=>i"      (infixr 60)
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    id  	::      "i=>i"
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    inj,surj,bij::      "[i,i]=>i"
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defs
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    (*composition of relations and functions; NOT Suppes's relative product*)
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    comp_def	"r O s == {xz : domain(s)*range(r) . 
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                  		EX x y z. xz=<x,z> & <x,y>:s & <y,z>:r}"
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    (*the identity function for A*)
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    id_def	"id(A) == (lam x:A. x)"
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    (*one-to-one functions from A to B*)
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    inj_def      "inj(A,B) == { f: A->B. ALL w:A. ALL x:A. f`w=f`x --> w=x}"
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    (*onto functions from A to B*)
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    surj_def	"surj(A,B) == { f: A->B . ALL y:B. EX x:A. f`x=y}"
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    (*one-to-one and onto functions*)
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    bij_def	"bij(A,B) == inj(A,B) Int surj(A,B)"
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end