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Theorem resdm 4828
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 3087 . 2 dom 𝐴 ⊆ dom 𝐴
2 relssres 4827 . 2 ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) → (𝐴 ↾ dom 𝐴) = 𝐴)
31, 2mpan2 421 1 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wss 3041  dom cdm 4509  cres 4511  Rel wrel 4514
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-dm 4519  df-res 4521
This theorem is referenced by:  resindm  4831  resdm2  4999  relresfld  5038  relcoi1  5040  funimaexg  5177  fnex  5610  dftpos2  6126  pmresg  6538  dif1en  6741
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