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Theorem resdm 5867
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 3940 . 2 dom 𝐴 ⊆ dom 𝐴
2 relssres 5863 . 2 ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) → (𝐴 ↾ dom 𝐴) = 𝐴)
31, 2mpan2 690 1 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wss 3884  dom cdm 5523  cres 5525  Rel wrel 5528
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-dm 5533  df-res 5535
This theorem is referenced by:  resindm  5871  resdm2  6059  relresfld  6099  fimadmfoALT  6580  fnex  6961  dftpos2  7896  tfrlem11  8011  tfrlem15  8015  tfrlem16  8016  pmresg  8421  domss2  8664  axdc3lem4  9868  gruima  10217  reldisjun  30369  funresdm1  30371  bnj1321  32407  funsseq  33119  nosupbnd2lem1  33323  nosupbnd2  33324  noetalem2  33326  noetalem3  33327  alrmomodm  35766  relbrcoss  35839  unidmqs  36041  releldmqs  36045  releldmqscoss  36047  seff  41000  sblpnf  41001  itcoval1  45064
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