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Theorem dmresi 5138
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 3313 . . 3  |-  A  C_  _V
2 dmi 5026 . . 3  |-  dom  _I  =  _V
31, 2sseqtr4i 3326 . 2  |-  A  C_  dom  _I
4 ssdmres 5110 . 2  |-  ( A 
C_  dom  _I  <->  dom  (  _I  |`  A )  =  A )
53, 4mpbi 200 1  |-  dom  (  _I  |`  A )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1649   _Vcvv 2901    C_ wss 3265    _I cid 4436   dom cdm 4820    |` cres 4822
This theorem is referenced by:  fnresi  5504  iordsmo  6557  hartogslem1  7446  dfac9  7951  hsmexlem5  8245  dirdm  14608  wilthlem2  20721  wilthlem3  20722  ausisusgra  21249  cusgraexilem2  21344  relexpdm  24916  filnetlem3  26102  filnetlem4  26103  islinds2  26954  lindsind2  26960  f1linds  26966
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-dm 4830  df-res 4832
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