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Theorem resdm 4904
 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 3148 . 2 dom 𝐴 ⊆ dom 𝐴
2 relssres 4903 . 2 ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) → (𝐴 ↾ dom 𝐴) = 𝐴)
31, 2mpan2 422 1 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1335   ⊆ wss 3102  dom cdm 4585   ↾ cres 4587  Rel wrel 4590 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4591  df-rel 4592  df-dm 4595  df-res 4597 This theorem is referenced by:  resindm  4907  resdm2  5075  relresfld  5114  relcoi1  5116  funimaexg  5253  fnex  5688  dftpos2  6205  pmresg  6618  dif1en  6821
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