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Theorem dmresi 4691
Description: The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
Assertion
Ref Expression
dmresi dom ( I ↾ 𝐴) = 𝐴

Proof of Theorem dmresi
StepHypRef Expression
1 ssv 3020 . . 3 𝐴 ⊆ V
2 dmi 4578 . . 3 dom I = V
31, 2sseqtr4i 3033 . 2 𝐴 ⊆ dom I
4 ssdmres 4661 . 2 (𝐴 ⊆ dom I ↔ dom ( I ↾ 𝐴) = 𝐴)
53, 4mpbi 143 1 dom ( I ↾ 𝐴) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1285  Vcvv 2602  wss 2974   I cid 4051  dom cdm 4371  cres 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-dm 4381  df-res 4383
This theorem is referenced by:  fnresi  5047  iordsmo  5946
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