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 ) )