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

Theorem relopab 4960
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 2404 . 2  |-  { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ph }
21relopabi 4959 1  |-  Rel  { <. x ,  y >.  |  ph }
Colors of variables: wff set class
Syntax hints:   {copab 4225   Rel wrel 4842
This theorem is referenced by:  opabid2  4963  inopab  4964  difopab  4965  dfres2  5152  cnvopab  5233  funopab  5445  relmptopab  6251  elopabi  6371  relmpt2opab  6388  shftfn  11843  eltopspOLD  16938  lgsquadlem3  21093  relfae  24551  linedegen  25981  opelopab3  26308  prtlem12  26606  dicvalrelN  31668  diclspsn  31677  dih1dimatlem  31812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-opab 4227  df-xp 4843  df-rel 4844
  Copyright terms: Public domain W3C validator