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Theorem iprc 4777
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. (Contributed by NM, 1-Jan-2007.)
Assertion
Ref Expression
iprc  |-  -.  _I  e.  _V

Proof of Theorem iprc
StepHypRef Expression
1 vprc 4030 . . 3  |-  -.  _V  e.  _V
2 dmi 4724 . . . 4  |-  dom  _I  =  _V
32eleq1i 2183 . . 3  |-  ( dom 
_I  e.  _V  <->  _V  e.  _V )
41, 3mtbir 645 . 2  |-  -.  dom  _I  e.  _V
5 dmexg 4773 . 2  |-  (  _I  e.  _V  ->  dom  _I  e.  _V )
64, 5mto 636 1  |-  -.  _I  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1465   _Vcvv 2660    _I cid 4180   dom cdm 4509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by: (None)
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