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Theorem iprc 6966
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 20809. (Contributed by NM, 1-Jan-2007.)
Assertion
Ref Expression
iprc ¬ I ∈ V

Proof of Theorem iprc
StepHypRef Expression
1 vprc 4715 . . 3 ¬ V ∈ V
2 dmi 5244 . . . 4 dom I = V
32eleq1i 2674 . . 3 (dom I ∈ V ↔ V ∈ V)
41, 3mtbir 311 . 2 ¬ dom I ∈ V
5 dmexg 6962 . 2 ( I ∈ V → dom I ∈ V)
64, 5mto 186 1 ¬ I ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 1975  Vcvv 3168   I cid 4934  dom cdm 5024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-dm 5034  df-rn 5035
This theorem is referenced by: (None)
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