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Mirrors > Home > MPE Home > Th. List > iprc | Structured version Visualization version GIF version |
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 23083. (Contributed by NM, 1-Jan-2007.) |
Ref | Expression |
---|---|
iprc | ⊢ ¬ I ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmi 5911 | . . 3 ⊢ dom I = V | |
2 | vprc 5305 | . . 3 ⊢ ¬ V ∈ V | |
3 | 1, 2 | eqneltri 2844 | . 2 ⊢ ¬ dom I ∈ V |
4 | dmexg 7887 | . 2 ⊢ ( I ∈ V → dom I ∈ V) | |
5 | 3, 4 | mto 196 | 1 ⊢ ¬ I ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2098 Vcvv 3466 I cid 5563 dom cdm 5666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-dm 5676 df-rn 5677 |
This theorem is referenced by: (None) |
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