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

Theorem relopab 5416
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 2765 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
21relopabi 5414 1 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  {copab 4871  Rel wrel 5282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-opab 4872  df-xp 5283  df-rel 5284
This theorem is referenced by:  opabid2  5420  inopab  5421  difopab  5422  dfres2  5630  cnvopab  5716  funopab  6103  relmptopab  7081  elopabi  7432  relmpt2opab  7461  shftfn  14098  cicer  16731  joindmss  17273  meetdmss  17287  lgsquadlem3  25398  perpln1  25896  perpln2  25897  fpwrelmapffslem  29956  fpwrelmap  29957  relfae  30757  vvdifopab  34459  inxprnres  34492  prtlem12  34823  dicvalrelN  37141  diclspsn  37150  dih1dimatlem  37285  rfovcnvf1od  38972
  Copyright terms: Public domain W3C validator