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 ) ) |
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 ) ) |