Metamath Proof Explorer

Theorem ecres

Description: Restricted coset of B . (Contributed by Peter Mazsa, 9-Dec-2018)

Ref Expression
Assertion ecres 
|- [ B ] ( R |` A ) = { x | ( B e. A /\ B R x ) }

Proof

Step Hyp Ref Expression
1 elecres 
 |-  ( x e. _V -> ( x e. [ B ] ( R |` A ) <-> ( B e. A /\ B R x ) ) )
2 1 elv 
 |-  ( x e. [ B ] ( R |` A ) <-> ( B e. A /\ B R x ) )
3 2 eqabi 
 |-  [ B ] ( R |` A ) = { x | ( B e. A /\ B R x ) }