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

Theorem resdm 6026
Description: A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
Assertion
Ref Expression
resdm (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)

Proof of Theorem resdm
StepHypRef Expression
1 ssid 4004 . 2 dom 𝐴 ⊆ dom 𝐴
2 relssres 6022 . 2 ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) → (𝐴 ↾ dom 𝐴) = 𝐴)
31, 2mpan2 689 1 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wss 3948  dom cdm 5676  cres 5678  Rel wrel 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-dm 5686  df-res 5688
This theorem is referenced by:  resindm  6030  reldisjun  6032  relresdm1  6033  resdm2  6230  relresfld  6275  fimadmfoALT  6816  fnex  7218  dftpos2  8227  tfrlem11  8387  tfrlem15  8391  tfrlem16  8392  pmresg  8863  domss2  9135  axdc3lem4  10447  gruima  10796  nosupbnd2lem1  27215  nosupbnd2  27216  noinfbnd2lem1  27230  noinfbnd2  27231  noetasuplem2  27234  noetasuplem3  27235  noetasuplem4  27236  noetainflem2  27238  bnj1321  34033  funsseq  34734  alrmomodm  37223  relbrcoss  37311  unidmqs  37519  releldmqs  37523  releldmqscoss  37525  seff  43058  sblpnf  43059  f1cof1blem  45774  funfocofob  45776  itcoval1  47339
  Copyright terms: Public domain W3C validator