Theorem rexima 5877

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 3219	. . . 4 |- ( ( B C_ A /\ y e. B ) -> y e. A )
2		funfvex 5643	. . . . 5 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
3	2	funfni 5422	. . . 4 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
4	1, 3	sylan2 286	. . 3 |- ( ( F Fn A /\ ( B C_ A /\ y e. B ) ) -> ( F ` y ) e. _V )
5	4	anassrs 400	. 2 |- ( ( ( F Fn A /\ B C_ A ) /\ y e. B ) -> ( F ` y ) e. _V )
6		fvelimab 5689	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( x e. ( F " B ) <-> E. y e. B ( F ` y ) = x ) )
7		eqcom 2231	. . . 4 |- ( ( F ` y ) = x <-> x = ( F ` y ) )
8	7	rexbii 2537	. . 3 |- ( E. y e. B ( F ` y ) = x <-> E. y e. B x = ( F ` y ) )
9	6, 8	bitrdi 196	. 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 277	. 2 |- ( ( ( F Fn A /\ B C_ A ) /\ x = ( F ` y ) ) -> ( ph <-> ps ) )
12	5, 9, 11	rexxfr2d 4555	1 |- ( ( F Fn A /\ B C_ A ) -> ( E. x e. ( F " B ) ph <-> E. y e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   _Vcvv 2799   C_ wss 3197   "cima 4721   Fn wfn 5312   ` cfv 5317

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325

This theorem is referenced by:  supisolem  7171