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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 4020 | . . 3 ⊢ ¬ V ∈ V | |
2 | dmi 4714 | . . . 4 ⊢ dom I = V | |
3 | 2 | eleq1i 2180 | . . 3 ⊢ (dom I ∈ V ↔ V ∈ V) |
4 | 1, 3 | mtbir 643 | . 2 ⊢ ¬ dom I ∈ V |
5 | dmexg 4761 | . 2 ⊢ ( I ∈ V → dom I ∈ V) | |
6 | 4, 5 | mto 634 | 1 ⊢ ¬ I ∈ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1463 Vcvv 2657 I cid 4170 dom cdm 4499 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-dm 4509 df-rn 4510 |
This theorem is referenced by: (None) |
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