Theorem pwvabrel 5716

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.)

Assertion

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

Proof of Theorem pwvabrel

Step	Hyp	Ref	Expression
1		pwvrel 5715	. . 3 ⊢ (𝑥 ∈ V → (𝑥 ∈ 𝒫 (V × V) ↔ Rel 𝑥))
2	1	elv 3467	. 2 ⊢ (𝑥 ∈ 𝒫 (V × V) ↔ Rel 𝑥)
3	2	eqabi 2905	1 ⊢ 𝒫 (V × V) = {𝑥 ∣ Rel 𝑥}

Syntax hints:   ↔ wb 209   = wceq 1568   ∈ wcel 2150  {cab 2748  Vcvv 3462  𝒫 cpw 4567   × cxp 5663  Rel wrel 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-v 3464  df-ss 3930  df-pw 4569  df-rel 5672

This theorem is referenced by: (None)