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

Theorem reldom 7575
Description: Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
Assertion
Ref Expression
reldom  |-  Rel  ~<_

Proof of Theorem reldom
Dummy variables  x  y  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dom 7571 . 2  |-  ~<_  =  { <. x ,  y >.  |  E. f  f : x -1-1-> y }
21relopabi 4971 1  |-  Rel  ~<_
Colors of variables: wff setvar class
Syntax hints:   E.wex 1659   Rel wrel 4851   -1-1->wf1 5590    ~<_ cdom 7567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-opab 4477  df-xp 4852  df-rel 4853  df-dom 7571
This theorem is referenced by:  relsdom  7576  brdomg  7579  brdomi  7580  domtr  7621  undom  7658  xpdom2  7665  xpdom1g  7667  domunsncan  7670  sbth  7690  sbthcl  7692  dom0  7698  fodomr  7721  pwdom  7722  domssex  7731  mapdom1  7735  mapdom2  7741  fineqv  7785  infsdomnn  7830  infn0  7831  elharval  8076  harword  8078  domwdom  8087  unxpwdom  8102  infdifsn  8159  infdiffi  8160  ac10ct  8461  iunfictbso  8541  cdadom1  8612  cdainf  8618  infcda1  8619  pwcdaidm  8621  cdalepw  8622  unctb  8631  infcdaabs  8632  infunabs  8633  infpss  8643  infmap2  8644  fictb  8671  infpssALT  8739  fin34  8816  ttukeylem1  8935  fodomb  8950  wdomac  8951  brdom3  8952  iundom2g  8961  iundom  8963  infxpidm  8983  iunctb  8995  gchdomtri  9050  pwfseq  9085  pwxpndom2  9086  pwxpndom  9087  pwcdandom  9088  gchpwdom  9091  gchaclem  9099  reexALT  11292  hashdomi  12552  cctop  19998  1stcrestlem  20444  2ndcdisj2  20449  dis2ndc  20452  hauspwdom  20493  ufilen  20922  ovoliunnul  22437  uniiccdif  22512  ovoliunnfl  31890  voliunnfl  31892  volsupnfl  31893  nnfoctb  37238  meadjiun  38034  caragenunicl  38075
  Copyright terms: Public domain W3C validator