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

Theorem relopab 5280
Description: A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) Removed DV restrictions. (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 2651 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
21relopabi 5278 1 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  {copab 4745  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150
This theorem is referenced by:  opabid2  5284  inopab  5285  difopab  5286  dfres2  5488  cnvopab  5568  funopab  5961  relmptopab  6925  elopabi  7276  relmpt2opab  7304  shftfn  13857  cicer  16513  joindmss  17054  meetdmss  17068  lgsquadlem3  25152  perpln1  25650  perpln2  25651  fpwrelmapffslem  29635  fpwrelmap  29636  relfae  30438  vvdifopab  34165  inxprnres  34201  prtlem12  34471  dicvalrelN  36791  diclspsn  36800  dih1dimatlem  36935  rfovcnvf1od  38615
  Copyright terms: Public domain W3C validator