Theorem cbvfo 5788

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 5442	. . 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 2180	. . . 4 |- ( y = ( F ` x ) -> ( ps <-> ph ) )
5	4	ralrn 5656	. . 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 5443	. . 3 |- ( F : A -onto-> B -> ran F = B )
8	7	raleqdv 2679	. 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 1353   A.wral 2455   ran crn 4629   Fn wfn 5213   -onto->wfo 5216   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224  df-fv 5226

This theorem is referenced by:  cocan2  5791  supisolem  7009