Metamath Proof Explorer


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 ) )