Theorem relmptopab

Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 7-Aug-2014) (Proof shortened by Mario Carneiro, 24-Dec-2016)

	Ref	Expression
Hypothesis	relmptopab.1	|- F = ( x e. A |-> { <. y , z >. | ph } )
Assertion	relmptopab	|- Rel ( F ` B )

Proof

Step	Hyp	Ref	Expression
1		relmptopab.1	|- F = ( x e. A |-> { <. y , z >. | ph } )
2	1	fvmptss	|- ( A. x e. A { <. y , z >. | ph } C_ ( _V X. _V ) -> ( F ` B ) C_ ( _V X. _V ) )
3		relopab	|- Rel { <. y , z >. | ph }
4		df-rel	|- ( Rel { <. y , z >. | ph } <-> { <. y , z >. | ph } C_ ( _V X. _V ) )
5	3 4	mpbi	|- { <. y , z >. | ph } C_ ( _V X. _V )
6	5	a1i	|- ( x e. A -> { <. y , z >. | ph } C_ ( _V X. _V ) )
7	2 6	mprg	|- ( F ` B ) C_ ( _V X. _V )
8		df-rel	|- ( Rel ( F ` B ) <-> ( F ` B ) C_ ( _V X. _V ) )
9	7 8	mpbir	|- Rel ( F ` B )