Metamath Proof Explorer


Theorem eleqvrelsrel

Description: For sets, being an element of the class of equivalence relations is equivalent to satisfying the equivalence relation predicate. (Contributed by Peter Mazsa, 24-Aug-2021)

Ref Expression
Assertion eleqvrelsrel ( 𝑅𝑉 → ( 𝑅 ∈ EqvRels ↔ EqvRel 𝑅 ) )

Proof

Step Hyp Ref Expression
1 elrelsrel ( 𝑅𝑉 → ( 𝑅 ∈ Rels ↔ Rel 𝑅 ) )
2 1 anbi2d ( 𝑅𝑉 → ( ( ( ( I ↾ dom 𝑅 ) ⊆ 𝑅 𝑅𝑅 ∧ ( 𝑅𝑅 ) ⊆ 𝑅 ) ∧ 𝑅 ∈ Rels ) ↔ ( ( ( I ↾ dom 𝑅 ) ⊆ 𝑅 𝑅𝑅 ∧ ( 𝑅𝑅 ) ⊆ 𝑅 ) ∧ Rel 𝑅 ) ) )
3 eleqvrels2 ( 𝑅 ∈ EqvRels ↔ ( ( ( I ↾ dom 𝑅 ) ⊆ 𝑅 𝑅𝑅 ∧ ( 𝑅𝑅 ) ⊆ 𝑅 ) ∧ 𝑅 ∈ Rels ) )
4 dfeqvrel2 ( EqvRel 𝑅 ↔ ( ( ( I ↾ dom 𝑅 ) ⊆ 𝑅 𝑅𝑅 ∧ ( 𝑅𝑅 ) ⊆ 𝑅 ) ∧ Rel 𝑅 ) )
5 2 3 4 3bitr4g ( 𝑅𝑉 → ( 𝑅 ∈ EqvRels ↔ EqvRel 𝑅 ) )