Theorem pwvabrel 5689

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 5688	. . 3 ⊢ (𝑥 ∈ V → (𝑥 ∈ 𝒫 (V × V) ↔ Rel 𝑥))
2	1	elv 3452	. 2 ⊢ (𝑥 ∈ 𝒫 (V × V) ↔ Rel 𝑥)
3	2	eqabi 2863	1 ⊢ 𝒫 (V × V) = {𝑥 ∣ Rel 𝑥}

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {cab 2707  Vcvv 3447  𝒫 cpw 4563   × cxp 5636  Rel wrel 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-ss 3931  df-pw 4565  df-rel 5645

This theorem is referenced by: (None)