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Theorem resdm 5429
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 3616 . 2 dom 𝐴 ⊆ dom 𝐴
2 relssres 5425 . 2 ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) → (𝐴 ↾ dom 𝐴) = 𝐴)
31, 2mpan2 706 1 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wss 3567  dom cdm 5104  cres 5106  Rel wrel 5109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-dm 5114  df-res 5116
This theorem is referenced by:  resindm  5432  resdm2  5612  relresfld  5650  fnex  6466  dftpos2  7354  tfrlem11  7469  tfrlem15  7473  tfrlem16  7474  pmresg  7870  domss2  8104  axdc3lem4  9260  gruima  9609  funresdm1  29388  bnj1321  31069  funsseq  31642  nosupbnd2lem1  31835  nosupbnd2  31836  noetalem2  31838  noetalem3  31839  seff  38328  sblpnf  38329
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