Theorem ecelqsw

Description: Membership of an equivalence class in a quotient set. More restrictive antecedent; kept for backward compatibility; for new work, prefer ecelqs . (Contributed by Jeff Madsen, 10-Jun-2010) (Revised by Mario Carneiro, 9-Jul-2014) (Proof shortened by AV, 25-Nov-2025)

	Ref	Expression
Assertion	ecelqsw	|- ( ( R e. V /\ B e. A ) -> [ B ] R e. ( A /. R ) )

Proof

Step	Hyp	Ref	Expression
1		resexg	|- ( R e. V -> ( R |` A ) e. _V )
2		ecelqs	|- ( ( ( R |` A ) e. _V /\ B e. A ) -> [ B ] R e. ( A /. R ) )
3	1 2	sylan	|- ( ( R e. V /\ B e. A ) -> [ B ] R e. ( A /. R ) )