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Metamath Proof Explorer

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 )