Theorem rexima 5706

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 3123	. . . 4 |- ( ( B C_ A /\ y e. B ) -> y e. A )
2		funfvex 5486	. . . . 5 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
3	2	funfni 5271	. . . 4 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
4	1, 3	sylan2 284	. . 3 |- ( ( F Fn A /\ ( B C_ A /\ y e. B ) ) -> ( F ` y ) e. _V )
5	4	anassrs 398	. 2 |- ( ( ( F Fn A /\ B C_ A ) /\ y e. B ) -> ( F ` y ) e. _V )
6		fvelimab 5525	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( x e. ( F " B ) <-> E. y e. B ( F ` y ) = x ) )
7		eqcom 2159	. . . 4 |- ( ( F ` y ) = x <-> x = ( F ` y ) )
8	7	rexbii 2464	. . 3 |- ( E. y e. B ( F ` y ) = x <-> E. y e. B x = ( F ` y ) )
9	6, 8	bitrdi 195	. 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 275	. 2 |- ( ( ( F Fn A /\ B C_ A ) /\ x = ( F ` y ) ) -> ( ph <-> ps ) )
12	5, 9, 11	rexxfr2d 4426	1 |- ( ( F Fn A /\ B C_ A ) -> ( E. x e. ( F " B ) ph <-> E. y e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   E.wrex 2436   _Vcvv 2712   C_ wss 3102   "cima 4590   Fn wfn 5166   ` cfv 5171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-fv 5179

This theorem is referenced by:  supisolem  6953