Theorem relopab 5767

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 2729	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2	1	relopabi 5765	1 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  {copab 5154  Rel wrel 5624

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-opab 5155  df-xp 5625  df-rel 5626

This theorem is referenced by:  relmptopab  7599  pwfir  9206  bj-0nelopab  37044  relcic  49034