Theorem relopab 5801

Description: A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) Remove disjoint variable conditions. (Revised by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)

Assertion

Ref	Expression
relopab	⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem relopab

Step	Hyp	Ref	Expression
1		eqid 2734	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2	1	relopabi 5799	1 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  {copab 5179  Rel wrel 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-opab 5180  df-xp 5658  df-rel 5659

This theorem is referenced by:  relmptopab  7652  pwfir  9322  bj-0nelopab  37013  relcic  48906