Theorem cnvelrels

Description: The converse of a set is an element of the class of relations. (Contributed by Peter Mazsa, 18-Aug-2019)

	Ref	Expression
Assertion	cnvelrels	⊢ ( 𝐴 ∈ 𝑉 → ◡ 𝐴 ∈ Rels )

Proof

Step	Hyp	Ref	Expression
1		relcnv	⊢ Rel ◡ 𝐴
2		cnvexg	⊢ ( 𝐴 ∈ 𝑉 → ◡ 𝐴 ∈ V )
3		elrelsrel	⊢ ( ◡ 𝐴 ∈ V → ( ◡ 𝐴 ∈ Rels ↔ Rel ◡ 𝐴 ) )
4	2 3	syl	⊢ ( 𝐴 ∈ 𝑉 → ( ◡ 𝐴 ∈ Rels ↔ Rel ◡ 𝐴 ) )
5	1 4	mpbiri	⊢ ( 𝐴 ∈ 𝑉 → ◡ 𝐴 ∈ Rels )