Theorem relopab 5790

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 2730	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2	1	relopabi 5788	1 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  {copab 5172  Rel wrel 5646

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-opab 5173  df-xp 5647  df-rel 5648

This theorem is referenced by:  relmptopab  7642  pwfir  9273  bj-0nelopab  37061  relcic  49038