Theorem pwvabrel 5629

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 5628	. . 3 ⊢ (𝑥 ∈ V → (𝑥 ∈ 𝒫 (V × V) ↔ Rel 𝑥))
2	1	elv 3428	. 2 ⊢ (𝑥 ∈ 𝒫 (V × V) ↔ Rel 𝑥)
3	2	abbi2i 2878	1 ⊢ 𝒫 (V × V) = {𝑥 ∣ Rel 𝑥}

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2108  {cab 2715  Vcvv 3422  𝒫 cpw 4530   × cxp 5578  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-pw 4532  df-rel 5587

This theorem is referenced by: (None)