Theorem eleqvrelsrel

Description: For sets, being an element of the class of equivalence relations is equivalent to satisfying the equivalence relation predicate. (Contributed by Peter Mazsa, 24-Aug-2021)

	Ref	Expression
Assertion	eleqvrelsrel	|- ( R e. V -> ( R e. EqvRels <-> EqvRel R ) )

Proof

Step	Hyp	Ref	Expression
1		elrelsrel	|- ( R e. V -> ( R e. Rels <-> Rel R ) )
2	1	anbi2d	|- ( R e. V -> ( ( ( ( _I |` dom R ) C_ R /\ `' R C_ R /\ ( R o. R ) C_ R ) /\ R e. Rels ) <-> ( ( ( _I |` dom R ) C_ R /\ `' R C_ R /\ ( R o. R ) C_ R ) /\ Rel R ) ) )
3		eleqvrels2	|- ( R e. EqvRels <-> ( ( ( _I |` dom R ) C_ R /\ `' R C_ R /\ ( R o. R ) C_ R ) /\ R e. Rels ) )
4		dfeqvrel2	|- ( EqvRel R <-> ( ( ( _I |` dom R ) C_ R /\ `' R C_ R /\ ( R o. R ) C_ R ) /\ Rel R ) )
5	2 3 4	3bitr4g	|- ( R e. V -> ( R e. EqvRels <-> EqvRel R ) )