Theorem ralima 5665

Description: Universal 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
ralima	|- ( ( F Fn A /\ B C_ A ) -> ( A. x e. ( F " B ) ph <-> A. 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 ralima

Step	Hyp	Ref	Expression
1		ssel2 3097	. . . 4 |- ( ( B C_ A /\ y e. B ) -> y e. A )
2		funfvex 5446	. . . . 5 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
3	2	funfni 5231	. . . 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 5485	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( x e. ( F " B ) <-> E. y e. B ( F ` y ) = x ) )
7		eqcom 2142	. . . 4 |- ( ( F ` y ) = x <-> x = ( F ` y ) )
8	7	rexbii 2445	. . 3 |- ( E. y e. B ( F ` y ) = x <-> E. y e. B x = ( F ` y ) )
9	6, 8	syl6bb 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	ralxfr2d 4393	1 |- ( ( F Fn A /\ B C_ A ) -> ( A. x e. ( F " B ) ph <-> A. y e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   _Vcvv 2689   C_ wss 3076   "cima 4550   Fn wfn 5126   ` cfv 5131

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139

This theorem is referenced by:  supisolem  6903  qtopbasss  12729