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Theorem iprc 4943
Description: The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set, as in idcn 16987. (Contributed by NM, 1-Jan-2007.)
Assertion
Ref Expression
iprc  |-  -.  _I  e.  _V

Proof of Theorem iprc
StepHypRef Expression
1 vprc 4152 . . 3  |-  -.  _V  e.  _V
2 dmi 4893 . . . 4  |-  dom  _I  =  _V
32eleq1i 2346 . . 3  |-  ( dom 
_I  e.  _V  <->  _V  e.  _V )
41, 3mtbir 290 . 2  |-  -.  dom  _I  e.  _V
5 dmexg 4939 . 2  |-  (  _I  e.  _V  ->  dom  _I  e.  _V )
64, 5mto 167 1  |-  -.  _I  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1684   _Vcvv 2788    _I cid 4304   dom cdm 4689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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