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	$$⊢A\in 𝒫\left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \left(A\in \mathrm{V}\wedge \mathrm{Rel}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢A\in 𝒫\left(\mathrm{V}\times \mathrm{V}\right)\to A\in \mathrm{V}$$
2		pwvrel	$$⊢A\in \mathrm{V}\to \left(A\in 𝒫\left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \mathrm{Rel}A\right)$$
3	1 2	biadanii	$$⊢A\in 𝒫\left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \left(A\in \mathrm{V}\wedge \mathrm{Rel}A\right)$$