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Metamath Proof Explorer

Theorem bj-pwvrelb

Description: Characterization of the elements of the powerclass of the cartesian square of the universal class: they are exactly the sets which are binary relations. (Contributed by BJ, 16-Dec-2023)

		Ref	Expression
	Assertion	bj-pwvrelb	⊢ ( 𝐴 ∈ 𝒫 ( V × V ) ↔ ( 𝐴 ∈ V ∧ Rel 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ 𝒫 ( V × V ) → 𝐴 ∈ V )
2		pwvrel	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 ( V × V ) ↔ Rel 𝐴 ) )
3	1 2	biadanii	⊢ ( 𝐴 ∈ 𝒫 ( V × V ) ↔ ( 𝐴 ∈ V ∧ Rel 𝐴 ) )