This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ⊢ R ∈ V → R ∈ Rels ↔ Rel ⁡ R

Proof

Step Hyp Ref Expression
1 elrels2 ⊢ R ∈ V → R ∈ Rels ↔ R ⊆ V × V
2 df-rel ⊢ Rel ⁡ R ↔ R ⊆ V × V
3 1 2 bitr4di ⊢ R ∈ V → R ∈ Rels ↔ Rel ⁡ R