Theorem cbvfo 5764

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 5422	. . 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 2173	. . . 4 |- ( y = ( F ` x ) -> ( ps <-> ph ) )
5	4	ralrn 5634	. . 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 5423	. . 3 |- ( F : A -onto-> B -> ran F = B )
8	7	raleqdv 2671	. 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 1348   A.wral 2448   ran crn 4612   Fn wfn 5193   -onto->wfo 5196   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206

This theorem is referenced by:  cocan2  5767  supisolem  6985