Theorem ralima

Description: Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015) Reduce DV conditions. (Revised by Matthew House, 14-Aug-2025)

	Ref	Expression
Hypothesis	ralima.x	|- ( x = ( F ` y ) -> ( ph <-> ps ) )
Assertion	ralima	|- ( ( F Fn A /\ B C_ A ) -> ( A. x e. ( F " B ) ph <-> A. y e. B ps ) )

Proof

Step	Hyp	Ref	Expression
1		ralima.x	|- ( x = ( F ` y ) -> ( ph <-> ps ) )
2		fnfun	|- ( F Fn A -> Fun F )
3	2	funfnd	|- ( F Fn A -> F Fn dom F )
4		fndm	|- ( F Fn A -> dom F = A )
5	4	sseq2d	|- ( F Fn A -> ( B C_ dom F <-> B C_ A ) )
6	5	biimpar	|- ( ( F Fn A /\ B C_ A ) -> B C_ dom F )
7		fvexd	|- ( ( ( F Fn dom F /\ B C_ dom F ) /\ y e. B ) -> ( F ` y ) e. _V )
8		fvelimab	|- ( ( F Fn dom F /\ B C_ dom F ) -> ( x e. ( F " B ) <-> E. y e. B ( F ` y ) = x ) )
9		eqcom	|- ( ( F ` y ) = x <-> x = ( F ` y ) )
10	9	rexbii	|- ( E. y e. B ( F ` y ) = x <-> E. y e. B x = ( F ` y ) )
11	8 10	bitrdi	|- ( ( F Fn dom F /\ B C_ dom F ) -> ( x e. ( F " B ) <-> E. y e. B x = ( F ` y ) ) )
12	1	adantl	|- ( ( ( F Fn dom F /\ B C_ dom F ) /\ x = ( F ` y ) ) -> ( ph <-> ps ) )
13	7 11 12	ralxfr2d	|- ( ( F Fn dom F /\ B C_ dom F ) -> ( A. x e. ( F " B ) ph <-> A. y e. B ps ) )
14	3 6 13	syl2an2r	|- ( ( F Fn A /\ B C_ A ) -> ( A. x e. ( F " B ) ph <-> A. y e. B ps ) )