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Theorem resdm 4677
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 2992 . 2 dom 𝐴 ⊆ dom 𝐴
2 relssres 4676 . 2 ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) → (𝐴 ↾ dom 𝐴) = 𝐴)
31, 2mpan2 409 1 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wss 2945  dom cdm 4373  cres 4375  Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-dm 4383  df-res 4385
This theorem is referenced by:  resdm2  4839  relresfld  4875  relcoi1  4877  funimaexg  5011  fnex  5411  dftpos2  5907  dif1en  6368
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