Theorem pwvrel 5595

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 4535	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 (V × V) ↔ 𝐴 ⊆ (V × V)))
2		df-rel 5555	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
3	1, 2	syl6bbr 291	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2113  Vcvv 3491   ⊆ wss 3929  𝒫 cpw 4532   × cxp 5546  Rel wrel 5553

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-in 3936  df-ss 3945  df-pw 4534  df-rel 5555

This theorem is referenced by:  pwvabrel  5596  bj-pwvrelb  34236