Theorem pwvrel 5682

Description: A set is a binary relation if and only if it belongs to the powerclass of the cartesian square of the universal class. (Contributed by Peter Mazsa, 14-Jun-2018.) (Revised by BJ, 16-Dec-2023.)

Assertion

Ref	Expression
pwvrel	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴))

Proof of Theorem pwvrel

Step	Hyp	Ref	Expression
1		elpwg 4559	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 (V × V) ↔ 𝐴 ⊆ (V × V)))
2		df-rel 5639	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
3	1, 2	bitr4di 289	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556   × cxp 5630  Rel wrel 5637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ss 3920  df-pw 4558  df-rel 5639

This theorem is referenced by:  pwvabrel  5683  bj-pwvrelb  37140