Theorem brab

Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999)

	Ref	Expression
Hypotheses	opelopab.1	|- A e. _V
	opelopab.2	|- B e. _V
	opelopab.3	|- ( x = A -> ( ph <-> ps ) )
	opelopab.4	|- ( y = B -> ( ps <-> ch ) )
	brab.5	|- R = { <. x , y >. | ph }
Assertion	brab	|- ( A R B <-> ch )

Proof

Step	Hyp	Ref	Expression
1		opelopab.1	|- A e. _V
2		opelopab.2	|- B e. _V
3		opelopab.3	|- ( x = A -> ( ph <-> ps ) )
4		opelopab.4	|- ( y = B -> ( ps <-> ch ) )
5		brab.5	|- R = { <. x , y >. | ph }
6	3 4 5	brabg	|- ( ( A e. _V /\ B e. _V ) -> ( A R B <-> ch ) )
7	1 2 6	mp2an	|- ( A R B <-> ch )