Theorem cbvfo 5753

Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Hypothesis

Ref	Expression
cbvfo.1	|- ( ( F ` x ) = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvfo	|- ( F : A -onto-> B -> ( A. x e. A ph <-> A. y e. B ps ) )

Distinct variable groups:   x, y, A   y, B   x, F, y   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)   B( x)

Proof of Theorem cbvfo

Step	Hyp	Ref	Expression
1		fofn 5412	. . 3 |- ( F : A -onto-> B -> F Fn A )
2		cbvfo.1	. . . . . 6 |- ( ( F ` x ) = y -> ( ph <-> ps ) )
3	2	bicomd 140	. . . . 5 |- ( ( F ` x ) = y -> ( ps <-> ph ) )
4	3	eqcoms 2168	. . . 4 |- ( y = ( F ` x ) -> ( ps <-> ph ) )
5	4	ralrn 5623	. . 3 |- ( F Fn A -> ( A. y e. ran F ps <-> A. x e. A ph ) )
6	1, 5	syl 14	. 2 |- ( F : A -onto-> B -> ( A. y e. ran F ps <-> A. x e. A ph ) )
7		forn 5413	. . 3 |- ( F : A -onto-> B -> ran F = B )
8	7	raleqdv 2667	. 2 |- ( F : A -onto-> B -> ( A. y e. ran F ps <-> A. y e. B ps ) )
9	6, 8	bitr3d 189	1 |- ( F : A -onto-> B -> ( A. x e. A ph <-> A. y e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   A.wral 2444   ran crn 4605   Fn wfn 5183   -onto->wfo 5186   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196

This theorem is referenced by:  cocan2  5756  supisolem  6973