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Metamath Proof Explorer

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 ) )