Theorem ralima 5589

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 3042	. . . 4 |- ( ( B C_ A /\ y e. B ) -> y e. A )
2		funfvex 5370	. . . . 5 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
3	2	funfni 5159	. . . 4 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
4	1, 3	sylan2 282	. . 3 |- ( ( F Fn A /\ ( B C_ A /\ y e. B ) ) -> ( F ` y ) e. _V )
5	4	anassrs 395	. 2 |- ( ( ( F Fn A /\ B C_ A ) /\ y e. B ) -> ( F ` y ) e. _V )
6		fvelimab 5409	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( x e. ( F " B ) <-> E. y e. B ( F ` y ) = x ) )
7		eqcom 2102	. . . 4 |- ( ( F ` y ) = x <-> x = ( F ` y ) )
8	7	rexbii 2401	. . 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 273	. 2 |- ( ( ( F Fn A /\ B C_ A ) /\ x = ( F ` y ) ) -> ( ph <-> ps ) )
12	5, 9, 11	ralxfr2d 4323	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 1299   e. wcel 1448   A.wral 2375   E.wrex 2376   _Vcvv 2641   C_ wss 3021   "cima 4480   Fn wfn 5054   ` cfv 5059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067

This theorem is referenced by:  supisolem  6810  qtopbasss  12443