Theorem braba

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

	Ref	Expression
Hypotheses	opelopaba.1	|- A e. _V
	opelopaba.2	|- B e. _V
	opelopaba.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
	braba.4	|- R = { <. x , y >. | ph }
Assertion	braba	|- ( A R B <-> ps )

Proof

Step	Hyp	Ref	Expression
1		opelopaba.1	|- A e. _V
2		opelopaba.2	|- B e. _V
3		opelopaba.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
4		braba.4	|- R = { <. x , y >. | ph }
5	3 4	brabga	|- ( ( A e. _V /\ B e. _V ) -> ( A R B <-> ps ) )
6	1 2 5	mp2an	|- ( A R B <-> ps )