Theorem pwvabrel

Description: The powerclass of the cartesian square of the universal class is the class of all sets which are binary relations. (Contributed by BJ, 21-Dec-2023)

	Ref	Expression
Assertion	pwvabrel	⊢ 𝒫 ( V × V ) = { 𝑥 ∣ Rel 𝑥 }

Proof

Step	Hyp	Ref	Expression
1		pwvrel	⊢ ( 𝑥 ∈ V → ( 𝑥 ∈ 𝒫 ( V × V ) ↔ Rel 𝑥 ) )
2	1	elv	⊢ ( 𝑥 ∈ 𝒫 ( V × V ) ↔ Rel 𝑥 )
3	2	abbi2i	⊢ 𝒫 ( V × V ) = { 𝑥 ∣ Rel 𝑥 }