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Theorem iprc 4628
 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 ∈ V

Proof of Theorem iprc
StepHypRef Expression
1 vprc 3916 . . 3 ¬ V ∈ V
2 dmi 4578 . . . 4 dom I = V
32eleq1i 2119 . . 3 (dom I ∈ V ↔ V ∈ V)
41, 3mtbir 606 . 2 ¬ dom I ∈ V
5 dmexg 4624 . 2 ( I ∈ V → dom I ∈ V)
64, 5mto 598 1 ¬ I ∈ V
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∈ wcel 1409  Vcvv 2574   I cid 4053  dom cdm 4373 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-dm 4383  df-rn 4384 This theorem is referenced by: (None)
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