Theorem brcnvin

Description: Intersection with a converse, binary relation. (Contributed by Peter Mazsa, 24-Mar-2024)

	Ref	Expression
Assertion	brcnvin	|- ( ( A e. V /\ B e. W ) -> ( A ( R i^i `' S ) B <-> ( A R B /\ B S A ) ) )

Proof

Step	Hyp	Ref	Expression
1		brin	|- ( A ( R i^i `' S ) B <-> ( A R B /\ A `' S B ) )
2		brcnvg	|- ( ( A e. V /\ B e. W ) -> ( A `' S B <-> B S A ) )
3	2	anbi2d	|- ( ( A e. V /\ B e. W ) -> ( ( A R B /\ A `' S B ) <-> ( A R B /\ B S A ) ) )
4	1 3	bitrid	|- ( ( A e. V /\ B e. W ) -> ( A ( R i^i `' S ) B <-> ( A R B /\ B S A ) ) )