Theorem brin

Description: The intersection of two relations. (Contributed by FL, 7-Oct-2008)

	Ref	Expression
Assertion	brin	|- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) )

Proof

Step	Hyp	Ref	Expression
1		elin	|- ( <. A , B >. e. ( R i^i S ) <-> ( <. A , B >. e. R /\ <. A , B >. e. S ) )
2		df-br	|- ( A ( R i^i S ) B <-> <. A , B >. e. ( R i^i S ) )
3		df-br	|- ( A R B <-> <. A , B >. e. R )
4		df-br	|- ( A S B <-> <. A , B >. e. S )
5	3 4	anbi12i	|- ( ( A R B /\ A S B ) <-> ( <. A , B >. e. R /\ <. A , B >. e. S ) )
6	1 2 5	3bitr4i	|- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) )