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Mirrors > Home > ILE Home > Th. List > cncfmptid | GIF version |
Description: The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.) |
Ref | Expression |
---|---|
cncfmptid | ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝑥) ∈ (𝑆–cn→𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr 3105 | . 2 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ) | |
2 | simpr 109 | . 2 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → 𝑇 ⊆ ℂ) | |
3 | simpll 518 | . . . . 5 ⊢ (((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ 𝑇) | |
4 | simpr 109 | . . . . 5 ⊢ (((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
5 | 3, 4 | sseldd 3098 | . . . 4 ⊢ (((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑇) |
6 | 5 | fmpttd 5575 | . . 3 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝑥):𝑆⟶𝑇) |
7 | simpr 109 | . . . 4 ⊢ ((𝑦 ∈ 𝑆 ∧ 𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+) | |
8 | 7 | a1i 9 | . . 3 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → ((𝑦 ∈ 𝑆 ∧ 𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+)) |
9 | eqid 2139 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝑆 ↦ 𝑥) = (𝑥 ∈ 𝑆 ↦ 𝑥) | |
10 | id 19 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
11 | simprll 526 | . . . . . . . 8 ⊢ (((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) ∧ ((𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ 𝑤 ∈ ℝ+)) → 𝑦 ∈ 𝑆) | |
12 | 9, 10, 11, 11 | fvmptd3 5514 | . . . . . . 7 ⊢ (((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) ∧ ((𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ 𝑤 ∈ ℝ+)) → ((𝑥 ∈ 𝑆 ↦ 𝑥)‘𝑦) = 𝑦) |
13 | id 19 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧) | |
14 | simprlr 527 | . . . . . . . 8 ⊢ (((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) ∧ ((𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ 𝑤 ∈ ℝ+)) → 𝑧 ∈ 𝑆) | |
15 | 9, 13, 14, 14 | fvmptd3 5514 | . . . . . . 7 ⊢ (((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) ∧ ((𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ 𝑤 ∈ ℝ+)) → ((𝑥 ∈ 𝑆 ↦ 𝑥)‘𝑧) = 𝑧) |
16 | 12, 15 | oveq12d 5792 | . . . . . 6 ⊢ (((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) ∧ ((𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ 𝑤 ∈ ℝ+)) → (((𝑥 ∈ 𝑆 ↦ 𝑥)‘𝑦) − ((𝑥 ∈ 𝑆 ↦ 𝑥)‘𝑧)) = (𝑦 − 𝑧)) |
17 | 16 | fveq2d 5425 | . . . . 5 ⊢ (((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) ∧ ((𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ 𝑤 ∈ ℝ+)) → (abs‘(((𝑥 ∈ 𝑆 ↦ 𝑥)‘𝑦) − ((𝑥 ∈ 𝑆 ↦ 𝑥)‘𝑧))) = (abs‘(𝑦 − 𝑧))) |
18 | 17 | breq1d 3939 | . . . 4 ⊢ (((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) ∧ ((𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ 𝑤 ∈ ℝ+)) → ((abs‘(((𝑥 ∈ 𝑆 ↦ 𝑥)‘𝑦) − ((𝑥 ∈ 𝑆 ↦ 𝑥)‘𝑧))) < 𝑤 ↔ (abs‘(𝑦 − 𝑧)) < 𝑤)) |
19 | 18 | exbiri 379 | . . 3 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (((𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ 𝑤 ∈ ℝ+) → ((abs‘(𝑦 − 𝑧)) < 𝑤 → (abs‘(((𝑥 ∈ 𝑆 ↦ 𝑥)‘𝑦) − ((𝑥 ∈ 𝑆 ↦ 𝑥)‘𝑧))) < 𝑤))) |
20 | 6, 8, 19 | elcncf1di 12735 | . 2 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → ((𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝑥) ∈ (𝑆–cn→𝑇))) |
21 | 1, 2, 20 | mp2and 429 | 1 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝑥) ∈ (𝑆–cn→𝑇)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 ⊆ wss 3071 class class class wbr 3929 ↦ cmpt 3989 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 < clt 7800 − cmin 7933 ℝ+crp 9441 abscabs 10769 –cn→ccncf 12726 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-map 6544 df-cncf 12727 |
This theorem is referenced by: expcncf 12761 dvcnp2cntop 12832 |
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