Metamath Proof Explorer


Theorem elrels2

Description: The element of the relations class ( df-rels ) and the relation predicate ( df-rel ) are the same when R is a set. (Contributed by Peter Mazsa, 14-Jun-2018)

Ref Expression
Assertion elrels2
|- ( R e. V -> ( R e. Rels <-> R C_ ( _V X. _V ) ) )

Proof

Step Hyp Ref Expression
1 df-rels
 |-  Rels = ~P ( _V X. _V )
2 1 eleq2i
 |-  ( R e. Rels <-> R e. ~P ( _V X. _V ) )
3 elpwg
 |-  ( R e. V -> ( R e. ~P ( _V X. _V ) <-> R C_ ( _V X. _V ) ) )
4 2 3 syl5bb
 |-  ( R e. V -> ( R e. Rels <-> R C_ ( _V X. _V ) ) )