Metamath Proof Explorer


Theorem ralima

Description: Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015)

Ref Expression
Hypothesis rexima.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 rexima.x
 |-  ( x = ( F ` y ) -> ( ph <-> ps ) )
2 fvexd
 |-  ( ( ( F Fn A /\ B C_ A ) /\ y e. B ) -> ( F ` y ) e. _V )
3 fvelimab
 |-  ( ( F Fn A /\ B C_ A ) -> ( x e. ( F " B ) <-> E. y e. B ( F ` y ) = x ) )
4 eqcom
 |-  ( ( F ` y ) = x <-> x = ( F ` y ) )
5 4 rexbii
 |-  ( E. y e. B ( F ` y ) = x <-> E. y e. B x = ( F ` y ) )
6 3 5 bitrdi
 |-  ( ( F Fn A /\ B C_ A ) -> ( x e. ( F " B ) <-> E. y e. B x = ( F ` y ) ) )
7 1 adantl
 |-  ( ( ( F Fn A /\ B C_ A ) /\ x = ( F ` y ) ) -> ( ph <-> ps ) )
8 2 6 7 ralxfr2d
 |-  ( ( F Fn A /\ B C_ A ) -> ( A. x e. ( F " B ) ph <-> A. y e. B ps ) )