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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 21109. (Contributed by NM, 1-Jan-2007.) |
Ref | Expression |
---|---|
iprc | ⊢ ¬ I ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vprc 4829 | . . 3 ⊢ ¬ V ∈ V | |
2 | dmi 5372 | . . . 4 ⊢ dom I = V | |
3 | 2 | eleq1i 2721 | . . 3 ⊢ (dom I ∈ V ↔ V ∈ V) |
4 | 1, 3 | mtbir 312 | . 2 ⊢ ¬ dom I ∈ V |
5 | dmexg 7139 | . 2 ⊢ ( I ∈ V → dom I ∈ V) | |
6 | 4, 5 | mto 188 | 1 ⊢ ¬ I ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2030 Vcvv 3231 I cid 5052 dom cdm 5143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-dm 5153 df-rn 5154 |
This theorem is referenced by: (None) |
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