Theorem cbvexfo 5426

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 5108	. . 3 |- ( F : A -onto-> B -> F Fn A )
2		cbvfo.1	. . . . . 6 |- ( ( F ` x ) = y -> ( ph <-> ps ) )
3	2	bicomd 129	. . . . 5 |- ( ( F ` x ) = y -> ( ps <-> ph ) )
4	3	eqcoms 2043	. . . 4 |- ( y = ( F ` x ) -> ( ps <-> ph ) )
5	4	rexrn 5304	. . 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 5109	. . 3 |- ( F : A -onto-> B -> ran F = B )
8	7	rexeqdv 2512	. 2 |- ( F : A -onto-> B -> ( E. y e. ran F ps <-> E. y e. B ps ) )
9	6, 8	bitr3d 179	1 |- ( F : A -onto-> B -> ( E. x e. A ph <-> E. y e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   E.wrex 2307   ran crn 4346   Fn wfn 4897   -onto->wfo 4900   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910

This theorem is referenced by: (None)