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Theorem dmresi 5187
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 3360 . . 3  |-  A  C_  _V
2 dmi 5075 . . 3  |-  dom  _I  =  _V
31, 2sseqtr4i 3373 . 2  |-  A  C_  dom  _I
4 ssdmres 5159 . 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 1652   _Vcvv 2948    C_ wss 3312    _I cid 4485   dom cdm 4869    |` cres 4871
This theorem is referenced by:  fnresi  5553  iordsmo  6610  hartogslem1  7500  dfac9  8005  hsmexlem5  8299  dirdm  14667  wilthlem2  20840  wilthlem3  20841  ausisusgra  21368  cusgraexilem2  21464  relexpdm  25123  filnetlem3  26346  filnetlem4  26347  islinds2  27198  lindsind2  27204  f1linds  27210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-dm 4879  df-res 4881
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