Theorem brabga

Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	opelopabga.1	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
	brabga.2	|- R = { <. x , y >. | ph }
Assertion	brabga	|- ( ( A e. V /\ B e. W ) -> ( A R B <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		opelopabga.1	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
2		brabga.2	|- R = { <. x , y >. | ph }
3		df-br	|- ( A R B <-> <. A , B >. e. R )
4	2	eleq2i	|- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ph } )
5	3 4	bitri	|- ( A R B <-> <. A , B >. e. { <. x , y >. | ph } )
6	1	opelopabga	|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) )
7	5 6	syl5bb	|- ( ( A e. V /\ B e. W ) -> ( A R B <-> ps ) )