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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 4150 | . . 3 ⊢ ¬ V ∈ V | |
2 | dmi 4860 | . . . 4 ⊢ dom I = V | |
3 | 2 | eleq1i 2255 | . . 3 ⊢ (dom I ∈ V ↔ V ∈ V) |
4 | 1, 3 | mtbir 672 | . 2 ⊢ ¬ dom I ∈ V |
5 | dmexg 4909 | . 2 ⊢ ( I ∈ V → dom I ∈ V) | |
6 | 4, 5 | mto 663 | 1 ⊢ ¬ I ∈ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2160 Vcvv 2752 I cid 4306 dom cdm 4644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-dm 4654 df-rn 4655 |
This theorem is referenced by: (None) |
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