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Mirrors > Home > ILE Home > Th. List > iprc | 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. (Contributed by NM, 1-Jan-2007.) |
Ref | Expression |
---|---|
iprc | ⊢ ¬ I ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vprc 4108 | . . 3 ⊢ ¬ V ∈ V | |
2 | dmi 4813 | . . . 4 ⊢ dom I = V | |
3 | 2 | eleq1i 2230 | . . 3 ⊢ (dom I ∈ V ↔ V ∈ V) |
4 | 1, 3 | mtbir 661 | . 2 ⊢ ¬ dom I ∈ V |
5 | dmexg 4862 | . 2 ⊢ ( I ∈ V → dom I ∈ V) | |
6 | 4, 5 | mto 652 | 1 ⊢ ¬ I ∈ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2135 Vcvv 2721 I cid 4260 dom cdm 4598 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-dm 4608 df-rn 4609 |
This theorem is referenced by: (None) |
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