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Metamath Proof Explorer

Theorem relmpoopab

Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015)

		Ref	Expression
	Hypothesis	relmpoopab.1	\|- F = ( x e. A , y e. B \|-> { <. z , w >. \| ph } )
	Assertion	relmpoopab	\|- Rel ( C F D )

Proof

Step	Hyp	Ref	Expression
1		relmpoopab.1	\|- F = ( x e. A , y e. B \|-> { <. z , w >. \| ph } )
2		relopabv	\|- Rel { <. z , w >. \| ph }
3		df-rel	\|- ( Rel { <. z , w >. \| ph } <-> { <. z , w >. \| ph } C_ ( _V X. _V ) )
4	2 3	mpbi	\|- { <. z , w >. \| ph } C_ ( _V X. _V )
5	4	rgen2w	\|- A. x e. A A. y e. B { <. z , w >. \| ph } C_ ( _V X. _V )
6	1	ovmptss	\|- ( A. x e. A A. y e. B { <. z , w >. \| ph } C_ ( _V X. _V ) -> ( C F D ) C_ ( _V X. _V ) )
7	5 6	ax-mp	\|- ( C F D ) C_ ( _V X. _V )
8		df-rel	\|- ( Rel ( C F D ) <-> ( C F D ) C_ ( _V X. _V ) )
9	7 8	mpbir	\|- Rel ( C F D )