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 )