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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 ) )