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	bitrid	|- ( R e. V -> ( R e. Rels <-> R C_ ( _V X. _V ) ) )