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	$$⊢𝒫\left(\mathrm{V}\times \mathrm{V}\right)=\left\{x|\mathrm{Rel}x\right\}$$

Proof

Step	Hyp	Ref	Expression
1		pwvrel	$$⊢x\in \mathrm{V}\to \left(x\in 𝒫\left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \mathrm{Rel}x\right)$$
2	1	elv	$$⊢x\in 𝒫\left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \mathrm{Rel}x$$
3	2	eqabi	$$⊢𝒫\left(\mathrm{V}\times \mathrm{V}\right)=\left\{x|\mathrm{Rel}x\right\}$$