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 ) ) |
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 ) ) |