Theorem eltrrelsrel

Description: For sets, being an element of the class of transitive relations is equivalent to satisfying the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021)

	Ref	Expression
Assertion	eltrrelsrel	|- ( R e. V -> ( R e. TrRels <-> TrRel R ) )

Proof

Step	Hyp	Ref	Expression
1		elrelsrel	|- ( R e. V -> ( R e. Rels <-> Rel R ) )
2	1	anbi2d	|- ( R e. V -> ( ( ( R o. R ) C_ R /\ R e. Rels ) <-> ( ( R o. R ) C_ R /\ Rel R ) ) )
3		eltrrels2	|- ( R e. TrRels <-> ( ( R o. R ) C_ R /\ R e. Rels ) )
4		dftrrel2	|- ( TrRel R <-> ( ( R o. R ) C_ R /\ Rel R ) )
5	2 3 4	3bitr4g	|- ( R e. V -> ( R e. TrRels <-> TrRel R ) )