Theorem cbvfo 6544

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 6117	. . 3 |- ( F : A -onto-> B -> F Fn A )
2		cbvfo.1	. . . . . 6 |- ( ( F ` x ) = y -> ( ph <-> ps ) )
3	2	bicomd 213	. . . . 5 |- ( ( F ` x ) = y -> ( ps <-> ph ) )
4	3	eqcoms 2630	. . . 4 |- ( y = ( F ` x ) -> ( ps <-> ph ) )
5	4	ralrn 6362	. . 3 |- ( F Fn A -> ( A. y e. ran F ps <-> A. x e. A ph ) )
6	1, 5	syl 17	. 2 |- ( F : A -onto-> B -> ( A. y e. ran F ps <-> A. x e. A ph ) )
7		forn 6118	. . 3 |- ( F : A -onto-> B -> ran F = B )
8	7	raleqdv 3144	. 2 |- ( F : A -onto-> B -> ( A. y e. ran F ps <-> A. y e. B ps ) )
9	6, 8	bitr3d 270	1 |- ( F : A -onto-> B -> ( A. x e. A ph <-> A. y e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   A.wral 2912   ran crn 5115   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896

This theorem is referenced by:  cbvexfo  6545  cocan2  6547  f1oweALT  7152  supisolem  8379  qtopeu  21519  deg1leb  23855  dchrelbas4  24968  cnpconn  31212  cocanfo  33512