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	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ EqvRels ↔ EqvRel 𝑅 ) )

Proof

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