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Metamath Proof Explorer

Theorem pwvrel

Description: A set is a binary relation if and only if it belongs to the powerclass of the cartesian square of the universal class. (Contributed by Peter Mazsa, 14-Jun-2018) (Revised by BJ, 16-Dec-2023)

Ref Expression
Assertion pwvrel ⊢ A ∈ V → A ∈ 𝒫 V × V ↔ Rel ⁡ A

Proof

Step Hyp Ref Expression
1 elpwg ⊢ A ∈ V → A ∈ 𝒫 V × V ↔ A ⊆ V × V
2 df-rel ⊢ Rel ⁡ A ↔ A ⊆ V × V
3 1 2 bitr4di ⊢ A ∈ V → A ∈ 𝒫 V × V ↔ Rel ⁡ A