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Theorem iprc 4689
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 3963 . . 3  |-  -.  _V  e.  _V
2 dmi 4639 . . . 4  |-  dom  _I  =  _V
32eleq1i 2153 . . 3  |-  ( dom 
_I  e.  _V  <->  _V  e.  _V )
41, 3mtbir 631 . 2  |-  -.  dom  _I  e.  _V
5 dmexg 4685 . 2  |-  (  _I  e.  _V  ->  dom  _I  e.  _V )
64, 5mto 623 1  |-  -.  _I  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1438   _Vcvv 2619    _I cid 4106   dom cdm 4428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by: (None)
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