Theorem relbrcnvg

Description: When R is a relation, the sethood assumptions on brcnv can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	relbrcnvg	|- ( Rel R -> ( A `' R B <-> B R A ) )

Proof

Step	Hyp	Ref	Expression
1		relcnv	|- Rel `' R
2	1	brrelex12i	|- ( A `' R B -> ( A e. _V /\ B e. _V ) )
3	2	a1i	|- ( Rel R -> ( A `' R B -> ( A e. _V /\ B e. _V ) ) )
4		brrelex12	|- ( ( Rel R /\ B R A ) -> ( B e. _V /\ A e. _V ) )
5	4	ancomd	|- ( ( Rel R /\ B R A ) -> ( A e. _V /\ B e. _V ) )
6	5	ex	|- ( Rel R -> ( B R A -> ( A e. _V /\ B e. _V ) ) )
7		brcnvg	|- ( ( A e. _V /\ B e. _V ) -> ( A `' R B <-> B R A ) )
8	7	a1i	|- ( Rel R -> ( ( A e. _V /\ B e. _V ) -> ( A `' R B <-> B R A ) ) )
9	3 6 8	pm5.21ndd	|- ( Rel R -> ( A `' R B <-> B R A ) )