Theorem rexima

Description: Existential 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	rexima	|- ( ( F Fn A /\ B C_ A ) -> ( E. x e. ( F " B ) ph <-> E. y e. B ps ) )

Proof

Step	Hyp	Ref	Expression
1		ralima.x	|- ( x = ( F ` y ) -> ( ph <-> ps ) )
2	1	notbid	|- ( x = ( F ` y ) -> ( -. ph <-> -. ps ) )
3	2	ralima	|- ( ( F Fn A /\ B C_ A ) -> ( A. x e. ( F " B ) -. ph <-> A. y e. B -. ps ) )
4	3	notbid	|- ( ( F Fn A /\ B C_ A ) -> ( -. A. x e. ( F " B ) -. ph <-> -. A. y e. B -. ps ) )
5		dfrex2	|- ( E. x e. ( F " B ) ph <-> -. A. x e. ( F " B ) -. ph )
6		dfrex2	|- ( E. y e. B ps <-> -. A. y e. B -. ps )
7	4 5 6	3bitr4g	|- ( ( F Fn A /\ B C_ A ) -> ( E. x e. ( F " B ) ph <-> E. y e. B ps ) )