Theorem elsymrels2

Description: Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021)

	Ref	Expression
Assertion	elsymrels2	⊢ ( 𝑅 ∈ SymRels ↔ ( ◡ 𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) )

Proof

Step	Hyp	Ref	Expression
1		dfsymrels2	⊢ SymRels = { 𝑟 ∈ Rels ∣ ◡ 𝑟 ⊆ 𝑟 }
2		cnveq	⊢ ( 𝑟 = 𝑅 → ◡ 𝑟 = ◡ 𝑅 )
3		id	⊢ ( 𝑟 = 𝑅 → 𝑟 = 𝑅 )
4	2 3	sseq12d	⊢ ( 𝑟 = 𝑅 → ( ◡ 𝑟 ⊆ 𝑟 ↔ ◡ 𝑅 ⊆ 𝑅 ) )
5	1 4	rabeqel	⊢ ( 𝑅 ∈ SymRels ↔ ( ◡ 𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) )