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Theorem dmresi 5158
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 3373 . . 3  |-  A  C_  _V
2 dmi 5050 . . 3  |-  dom  _I  =  _V
31, 2sseqtr4i 3386 . 2  |-  A  C_  dom  _I
4 ssdmres 5129 . 2  |-  ( A 
C_  dom  _I  <->  dom  (  _I  |`  A )  =  A )
53, 4mpbi 208 1  |-  dom  (  _I  |`  A )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1364   _Vcvv 2970    C_ wss 3325    _I cid 4627   dom cdm 4836    |` cres 4838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-dm 4846  df-res 4848
This theorem is referenced by:  fnresi  5525  iordsmo  6814  hartogslem1  7752  dfac9  8301  hsmexlem5  8595  dirdm  15400  islinds2  18142  lindsind2  18148  f1linds  18154  wilthlem3  22351  ausisusgra  23198  cusgraexilem2  23294  relexpdm  27250  filnetlem3  28510  filnetlem4  28511
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