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 ) |
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 ) |