Theorem relelec

Description: Membership in an equivalence class when R is a relation. (Contributed by Mario Carneiro, 11-Sep-2015)

	Ref	Expression
Assertion	relelec	|- ( Rel R -> ( A e. [ B ] R <-> B R A ) )

Proof

Step	Hyp	Ref	Expression
1		elex	|- ( A e. [ B ] R -> A e. _V )
2		ecexr	|- ( A e. [ B ] R -> B e. _V )
3	1 2	jca	|- ( A e. [ B ] R -> ( A e. _V /\ B e. _V ) )
4	3	adantl	|- ( ( Rel R /\ A e. [ B ] R ) -> ( A e. _V /\ B e. _V ) )
5		brrelex12	|- ( ( Rel R /\ B R A ) -> ( B e. _V /\ A e. _V ) )
6	5	ancomd	|- ( ( Rel R /\ B R A ) -> ( A e. _V /\ B e. _V ) )
7		elecg	|- ( ( A e. _V /\ B e. _V ) -> ( A e. [ B ] R <-> B R A ) )
8	4 6 7	pm5.21nd	|- ( Rel R -> ( A e. [ B ] R <-> B R A ) )