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Metamath Proof Explorer

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 )