Theorem elsymrelsrel

Description: For sets, being an element of the class of symmetric relations ( df-symrels ) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021)

	Ref	Expression
Assertion	elsymrelsrel	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ SymRels ↔ SymRel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		elrelsrel	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ Rels ↔ Rel 𝑅 ) )
2	1	anbi2d	⊢ ( 𝑅 ∈ 𝑉 → ( ( ◡ 𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) ↔ ( ◡ 𝑅 ⊆ 𝑅 ∧ Rel 𝑅 ) ) )
3		elsymrels2	⊢ ( 𝑅 ∈ SymRels ↔ ( ◡ 𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) )
4		dfsymrel2	⊢ ( SymRel 𝑅 ↔ ( ◡ 𝑅 ⊆ 𝑅 ∧ Rel 𝑅 ) )
5	2 3 4	3bitr4g	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ SymRels ↔ SymRel 𝑅 ) )