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Metamath Proof Explorer

Theorem funimassov

Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013)

Ref Expression
Assertion funimassov 
|- ( ( Fun F /\ ( A X. B ) C_ dom F ) -> ( ( F " ( A X. B ) ) C_ C <-> A. x e. A A. y e. B ( x F y ) e. C ) )

Proof

Step Hyp Ref Expression
1 funimass4 
 |-  ( ( Fun F /\ ( A X. B ) C_ dom F ) -> ( ( F " ( A X. B ) ) C_ C <-> A. z e. ( A X. B ) ( F ` z ) e. C ) )
2 fveq2 
 |-  ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3 df-ov 
 |-  ( x F y ) = ( F ` <. x , y >. )
4 2 3 eqtr4di 
 |-  ( z = <. x , y >. -> ( F ` z ) = ( x F y ) )
5 4 eleq1d 
 |-  ( z = <. x , y >. -> ( ( F ` z ) e. C <-> ( x F y ) e. C ) )
6 5 ralxp 
 |-  ( A. z e. ( A X. B ) ( F ` z ) e. C <-> A. x e. A A. y e. B ( x F y ) e. C )
7 1 6 bitrdi 
 |-  ( ( Fun F /\ ( A X. B ) C_ dom F ) -> ( ( F " ( A X. B ) ) C_ C <-> A. x e. A A. y e. B ( x F y ) e. C ) )