Metamath Proof Explorer


Theorem elrelsrel

Description: The element of the relations class ( df-rels ) and the relation predicate are the same when R is a set. (Contributed by Peter Mazsa, 24-Nov-2018)

Ref Expression
Assertion elrelsrel
|- ( R e. V -> ( R e. Rels <-> Rel R ) )

Proof

Step Hyp Ref Expression
1 elrels2
 |-  ( R e. V -> ( R e. Rels <-> R C_ ( _V X. _V ) ) )
2 df-rel
 |-  ( Rel R <-> R C_ ( _V X. _V ) )
3 1 2 bitr4di
 |-  ( R e. V -> ( R e. Rels <-> Rel R ) )