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Theorem iprc 6993
 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 20871. (Contributed by NM, 1-Jan-2007.)
Assertion
Ref Expression
iprc ¬ I ∈ V

Proof of Theorem iprc
StepHypRef Expression
1 vprc 4724 . . 3 ¬ V ∈ V
2 dmi 5261 . . . 4 dom I = V
32eleq1i 2679 . . 3 (dom I ∈ V ↔ V ∈ V)
41, 3mtbir 312 . 2 ¬ dom I ∈ V
5 dmexg 6989 . 2 ( I ∈ V → dom I ∈ V)
64, 5mto 187 1 ¬ I ∈ V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 1977  Vcvv 3173   I cid 4948  dom cdm 5038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by: (None)
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