Theorem pwvrel 5715

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 4596	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 (V × V) ↔ 𝐴 ⊆ (V × V)))
2		df-rel 5673	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
3	1, 2	bitr4di 288	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2106  Vcvv 3470   ⊆ wss 3941  𝒫 cpw 4593   × cxp 5664  Rel wrel 5671

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3472  df-in 3948  df-ss 3958  df-pw 4595  df-rel 5673

This theorem is referenced by:  pwvabrel  5716  bj-pwvrelb  35566