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Theorem dmresi 4947
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 3170 . . 3  |-  A  C_  _V
2 dmi 4827 . . 3  |-  dom  _I  =  _V
31, 2sseqtrri 3183 . 2  |-  A  C_  dom  _I
4 ssdmres 4914 . 2  |-  ( A 
C_  dom  _I  <->  dom  (  _I  |`  A )  =  A )
53, 4mpbi 144 1  |-  dom  (  _I  |`  A )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1349   _Vcvv 2731    C_ wss 3122    _I cid 4274   dom cdm 4612    |` cres 4614
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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195
This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-v 2733  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-br 3991  df-opab 4052  df-id 4279  df-xp 4618  df-rel 4619  df-dm 4622  df-res 4624
This theorem is referenced by:  fnresi  5317  iordsmo  6280
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