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Theorem dmresi 5231
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 3357 . . 3  |-  A  C_  _V
2 dmi 5119 . . 3  |-  dom  _I  =  _V
31, 2sseqtr4i 3370 . 2  |-  A  C_  dom  _I
4 ssdmres 5203 . 2  |-  ( A 
C_  dom  _I  <->  dom  (  _I  |`  A )  =  A )
53, 4mpbi 201 1  |-  dom  (  _I  |`  A )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1654   _Vcvv 2965    C_ wss 3309    _I cid 4528   dom cdm 4913    |` cres 4915
This theorem is referenced by:  fnresi  5597  iordsmo  6655  hartogslem1  7547  dfac9  8054  hsmexlem5  8348  dirdm  14717  wilthlem2  20890  wilthlem3  20891  ausisusgra  21418  cusgraexilem2  21514  relexpdm  25170  filnetlem3  26451  filnetlem4  26452  islinds2  27372  lindsind2  27378  f1linds  27384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-br 4244  df-opab 4298  df-id 4533  df-xp 4919  df-rel 4920  df-dm 4923  df-res 4925
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