Metamath Proof Explorer


Theorem elrelsrel

Description: The element of the relations class ( df-rels ) and the relation predicate are the same when R is a set. (Contributed by Peter Mazsa, 24-Nov-2018)

Ref Expression
Assertion elrelsrel ( 𝑅𝑉 → ( 𝑅 ∈ Rels ↔ Rel 𝑅 ) )

Proof

Step Hyp Ref Expression
1 elrels2 ( 𝑅𝑉 → ( 𝑅 ∈ Rels ↔ 𝑅 ⊆ ( V × V ) ) )
2 df-rel ( Rel 𝑅𝑅 ⊆ ( V × V ) )
3 1 2 bitr4di ( 𝑅𝑉 → ( 𝑅 ∈ Rels ↔ Rel 𝑅 ) )