Theorem pwvrel 5701

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 5658	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
3	1, 2	bitr4di 292	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896  𝒫 cpw 4556   × cxp 5649  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ss 3897  df-pw 4558  df-rel 5658

This theorem is used by:  pwvabrel  5702  bj-pwvrelb  37906