ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iprc Unicode version

Theorem iprc 4856
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 4098 . . 3  |-  -.  _V  e.  _V
2 dmi 4803 . . . 4  |-  dom  _I  =  _V
32eleq1i 2223 . . 3  |-  ( dom 
_I  e.  _V  <->  _V  e.  _V )
41, 3mtbir 661 . 2  |-  -.  dom  _I  e.  _V
5 dmexg 4852 . 2  |-  (  _I  e.  _V  ->  dom  _I  e.  _V )
64, 5mto 652 1  |-  -.  _I  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 2128   _Vcvv 2712    _I cid 4250   dom cdm 4588
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-dm 4598  df-rn 4599
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator