Theorem cbvfo 5832

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 5482	. . 3 |- ( F : A -onto-> B -> F Fn A )
2		cbvfo.1	. . . . . 6 |- ( ( F ` x ) = y -> ( ph <-> ps ) )
3	2	bicomd 141	. . . . 5 |- ( ( F ` x ) = y -> ( ps <-> ph ) )
4	3	eqcoms 2199	. . . 4 |- ( y = ( F ` x ) -> ( ps <-> ph ) )
5	4	ralrn 5700	. . 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 5483	. . 3 |- ( F : A -onto-> B -> ran F = B )
8	7	raleqdv 2699	. 2 |- ( F : A -onto-> B -> ( A. y e. ran F ps <-> A. y e. B ps ) )
9	6, 8	bitr3d 190	1 |- ( F : A -onto-> B -> ( A. x e. A ph <-> A. y e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   A.wral 2475   ran crn 4664   Fn wfn 5253   -onto->wfo 5256   ` cfv 5258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fo 5264  df-fv 5266

This theorem is referenced by:  cocan2  5835  supisolem  7074