Metamath Proof Explorer


Theorem ecres2

Description: The restricted coset of B when B is an element of the restriction. (Contributed by Peter Mazsa, 16-Oct-2018)

Ref Expression
Assertion ecres2
|- ( B e. A -> [ B ] ( R |` A ) = [ B ] R )

Proof

Step Hyp Ref Expression
1 elecres
 |-  ( y e. _V -> ( y e. [ B ] ( R |` A ) <-> ( B e. A /\ B R y ) ) )
2 1 elv
 |-  ( y e. [ B ] ( R |` A ) <-> ( B e. A /\ B R y ) )
3 2 baib
 |-  ( B e. A -> ( y e. [ B ] ( R |` A ) <-> B R y ) )
4 3 abbi2dv
 |-  ( B e. A -> [ B ] ( R |` A ) = { y | B R y } )
5 dfec2
 |-  ( B e. A -> [ B ] R = { y | B R y } )
6 4 5 eqtr4d
 |-  ( B e. A -> [ B ] ( R |` A ) = [ B ] R )