Theorem inelv 3346

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 7705.

Assertion

Ref	Expression
inelv	|- -. I e. V

Proof of Theorem inelv

Step	Hyp	Ref	Expression
1		nvelv 2703	. . 3 |- -. V e. V
2		dmi 3315	. . . 4 |- dom I = V
3	2	eleq1i 1529	. . 3 |- (dom I e. V <-> V e. V)
4	1, 3	mtbir 192	. 2 |- -. dom I e. V
5		dmexg 3344	. 2 |- (I e. V -> dom I e. V)
6	4, 5	mto 106	1 |- -. I e. V

Syntax hints:  -. wn 2   e. wcel 955  Vcvv 1802  Icid 2820  dom cdm 3160

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-dm 3178  df-rn 3179

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