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Theorem resdm 6044
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 4006 . 2 dom 𝐴 ⊆ dom 𝐴
2 relssres 6040 . 2 ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) → (𝐴 ↾ dom 𝐴) = 𝐴)
31, 2mpan2 691 1 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wss 3951  dom cdm 5685  cres 5687  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dm 5695  df-res 5697
This theorem is referenced by:  resindm  6048  reldisjun  6050  relresdm1  6051  imadifssran  6171  resdm2  6251  relresfld  6296  fimadmfoALT  6831  fnex  7237  dftpos2  8268  tfrlem11  8428  tfrlem15  8432  tfrlem16  8433  pmresg  8910  domss2  9176  axdc3lem4  10493  gruima  10842  nosupbnd2lem1  27760  nosupbnd2  27761  noinfbnd2lem1  27775  noinfbnd2  27776  noetasuplem2  27779  noetasuplem3  27780  noetasuplem4  27781  noetainflem2  27783  bnj1321  35041  funsseq  35768  alrmomodm  38360  relbrcoss  38447  unidmqs  38655  releldmqs  38659  releldmqscoss  38661  seff  44328  sblpnf  44329  f1cof1blem  47086  funfocofob  47090  itcoval1  48584
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