Theorem rexima 5415

Description: Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Hypothesis

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

Assertion

Ref	Expression
rexima	|- ( ( F Fn A /\ B C_ A ) -> ( E. x e. ( F " B ) ph <-> E. y e. B ps ) )

Distinct variable groups:   ph, y   ps, x   x, F, y   x, B, y   x, A, y

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem rexima

Step	Hyp	Ref	Expression
1		ssel2 2994	. . . 4 |- ( ( B C_ A /\ y e. B ) -> y e. A )
2		funfvex 5212	. . . . 5 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
3	2	funfni 5019	. . . 4 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
4	1, 3	sylan2 280	. . 3 |- ( ( F Fn A /\ ( B C_ A /\ y e. B ) ) -> ( F ` y ) e. _V )
5	4	anassrs 392	. 2 |- ( ( ( F Fn A /\ B C_ A ) /\ y e. B ) -> ( F ` y ) e. _V )
6		fvelimab 5250	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( x e. ( F " B ) <-> E. y e. B ( F ` y ) = x ) )
7		eqcom 2083	. . . 4 |- ( ( F ` y ) = x <-> x = ( F ` y ) )
8	7	rexbii 2373	. . 3 |- ( E. y e. B ( F ` y ) = x <-> E. y e. B x = ( F ` y ) )
9	6, 8	syl6bb 194	. 2 |- ( ( F Fn A /\ B C_ A ) -> ( x e. ( F " B ) <-> E. y e. B x = ( F ` y ) ) )
10		rexima.x	. . 3 |- ( x = ( F ` y ) -> ( ph <-> ps ) )
11	10	adantl 271	. 2 |- ( ( ( F Fn A /\ B C_ A ) /\ x = ( F ` y ) ) -> ( ph <-> ps ) )
12	5, 9, 11	rexxfr2d 4215	1 |- ( ( F Fn A /\ B C_ A ) -> ( E. x e. ( F " B ) ph <-> E. y e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   E.wrex 2349   _Vcvv 2601   C_ wss 2973   "cima 4366   Fn wfn 4917   ` 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-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by:  supisolem  6421