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Theorem resdm 6027
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 4005 . 2 dom 𝐴 ⊆ dom 𝐴
2 relssres 6023 . 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 1542  wss 3949  dom cdm 5677  cres 5679  Rel wrel 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-dm 5687  df-res 5689
This theorem is referenced by:  resindm  6031  reldisjun  6033  relresdm1  6034  resdm2  6231  relresfld  6276  fimadmfoALT  6817  fnex  7219  dftpos2  8228  tfrlem11  8388  tfrlem15  8392  tfrlem16  8393  pmresg  8864  domss2  9136  axdc3lem4  10448  gruima  10797  nosupbnd2lem1  27218  nosupbnd2  27219  noinfbnd2lem1  27233  noinfbnd2  27234  noetasuplem2  27237  noetasuplem3  27238  noetasuplem4  27239  noetainflem2  27241  bnj1321  34038  funsseq  34739  alrmomodm  37228  relbrcoss  37316  unidmqs  37524  releldmqs  37528  releldmqscoss  37530  seff  43068  sblpnf  43069  f1cof1blem  45784  funfocofob  45786  itcoval1  47349
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