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

Theorem relopab 5004
Description: A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) (Unnecessary distinct variable restrictions were removed by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)
Assertion
Ref Expression
relopab  |-  Rel  { <. x ,  y >.  |  ph }

Proof of Theorem relopab
StepHypRef Expression
1 eqid 2438 . 2  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ph }
21relopabi 5003 1  |-  Rel  { <. x ,  y >.  |  ph }
Colors of variables: wff set class
Syntax hints:   {copab 4268   Rel wrel 4886
This theorem is referenced by:  opabid2  5007  inopab  5008  difopab  5009  dfres2  5196  cnvopab  5277  funopab  5489  relmptopab  6295  elopabi  6415  relmpt2opab  6432  shftfn  11893  eltopspOLD  16988  lgsquadlem3  21145  relfae  24603  linedegen  26082  opelopab3  26432  prtlem12  26730  dicvalrelN  32057  diclspsn  32066  dih1dimatlem  32201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4270  df-xp 4887  df-rel 4888
  Copyright terms: Public domain W3C validator