Description: Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011)
Ref | Expression | ||
---|---|---|---|
Hypothesis | opres.1 | |- B e. _V |
|
Assertion | opres | |- ( A e. D -> ( <. A , B >. e. ( C |` D ) <-> <. A , B >. e. C ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opres.1 | |- B e. _V |
|
2 | 1 | opelresi | |- ( <. A , B >. e. ( C |` D ) <-> ( A e. D /\ <. A , B >. e. C ) ) |
3 | 2 | baib | |- ( A e. D -> ( <. A , B >. e. ( C |` D ) <-> <. A , B >. e. C ) ) |