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Theorem iprc 5163
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 17352. (Contributed by NM, 1-Jan-2007.)
Assertion
Ref Expression
iprc  |-  -.  _I  e.  _V

Proof of Theorem iprc
StepHypRef Expression
1 vprc 4370 . . 3  |-  -.  _V  e.  _V
2 dmi 5113 . . . 4  |-  dom  _I  =  _V
32eleq1i 2505 . . 3  |-  ( dom 
_I  e.  _V  <->  _V  e.  _V )
41, 3mtbir 292 . 2  |-  -.  dom  _I  e.  _V
5 dmexg 5159 . 2  |-  (  _I  e.  _V  ->  dom  _I  e.  _V )
64, 5mto 170 1  |-  -.  _I  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1727   _Vcvv 2962    _I cid 4522   dom cdm 4907
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 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-un 4730
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 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-dm 4917  df-rn 4918
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