Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  relopab Structured version   Visualization version   GIF version

Theorem relopab 5694
 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
StepHypRef Expression
1 eqid 2825 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
21relopabi 5692 1 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 Colors of variables: wff setvar class Syntax hints:  {copab 5124  Rel wrel 5558 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-opab 5125  df-xp 5559  df-rel 5560 This theorem is referenced by:  opabid2  5698  inopab  5699  difopab  5700  dfres2  5907  cnvopab  5994  funopab  6386  relmptopab  7388  elopabi  7754  relmpoopab  7783  shftfn  14425  cicer  17069  joindmss  17610  meetdmss  17624  lgsquadlem3  25875  tgjustf  26176  perpln1  26413  perpln2  26414  fpwrelmapffslem  30384  fpwrelmap  30385  relfae  31395  satfrel  32501  bj-0nelopab  34242  vvdifopab  35392  inxprnres  35420  prtlem12  35873  dicvalrelN  38191  diclspsn  38200  dih1dimatlem  38335  rfovcnvf1od  40218
 Copyright terms: Public domain W3C validator