Theorem cbvexfo 5446

Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)

Hypothesis

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

Assertion

Ref	Expression
cbvexfo	|- ( F : A -onto-> B -> ( E. x e. A ph <-> E. 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 cbvexfo

Step	Hyp	Ref	Expression
1		fofn 5128	. . 3 |- ( F : A -onto-> B -> F Fn A )
2		cbvfo.1	. . . . . 6 |- ( ( F ` x ) = y -> ( ph <-> ps ) )
3	2	bicomd 139	. . . . 5 |- ( ( F ` x ) = y -> ( ps <-> ph ) )
4	3	eqcoms 2084	. . . 4 |- ( y = ( F ` x ) -> ( ps <-> ph ) )
5	4	rexrn 5325	. . 3 |- ( F Fn A -> ( E. y e. ran F ps <-> E. x e. A ph ) )
6	1, 5	syl 14	. 2 |- ( F : A -onto-> B -> ( E. y e. ran F ps <-> E. x e. A ph ) )
7		forn 5129	. . 3 |- ( F : A -onto-> B -> ran F = B )
8	7	rexeqdv 2556	. 2 |- ( F : A -onto-> B -> ( E. y e. ran F ps <-> E. y e. B ps ) )
9	6, 8	bitr3d 188	1 |- ( F : A -onto-> B -> ( E. x e. A ph <-> E. y e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   E.wrex 2349   ran crn 4364   Fn wfn 4917   -onto->wfo 4920   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930

This theorem is referenced by: (None)