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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcnre | Structured version Visualization version GIF version |
Description: A function continuous with respect to the standard topology, is a real mapping. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
fcnre.1 | ⊢ 𝐾 = (topGen‘ran (,)) |
fcnre.3 | ⊢ 𝑇 = ∪ 𝐽 |
sfcnre.5 | ⊢ 𝐶 = (𝐽 Cn 𝐾) |
fcnre.6 | ⊢ (𝜑 → 𝐹 ∈ 𝐶) |
Ref | Expression |
---|---|
fcnre | ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcnre.6 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐶) | |
2 | sfcnre.5 | . . . . 5 ⊢ 𝐶 = (𝐽 Cn 𝐾) | |
3 | 1, 2 | eleqtrdi 2900 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
4 | cntop1 21845 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
6 | fcnre.3 | . . . 4 ⊢ 𝑇 = ∪ 𝐽 | |
7 | 6 | toptopon 21522 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑇)) |
8 | 5, 7 | sylib 221 | . 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑇)) |
9 | fcnre.1 | . . . 4 ⊢ 𝐾 = (topGen‘ran (,)) | |
10 | retopon 23369 | . . . 4 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
11 | 9, 10 | eqeltri 2886 | . . 3 ⊢ 𝐾 ∈ (TopOn‘ℝ) |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℝ)) |
13 | cnf2 21854 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑇) ∧ 𝐾 ∈ (TopOn‘ℝ) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑇⟶ℝ) | |
14 | 8, 12, 3, 13 | syl3anc 1368 | 1 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∪ cuni 4800 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 (,)cioo 12726 topGenctg 16703 Topctop 21498 TopOnctopon 21515 Cn ccn 21829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ioo 12730 df-topgen 16709 df-top 21499 df-topon 21516 df-bases 21551 df-cn 21832 |
This theorem is referenced by: rfcnpre2 41660 cncmpmax 41661 rfcnpre3 41662 rfcnpre4 41663 rfcnnnub 41665 stoweidlem28 42670 stoweidlem29 42671 stoweidlem36 42678 stoweidlem43 42685 stoweidlem44 42686 stoweidlem47 42689 stoweidlem52 42694 stoweidlem57 42699 stoweidlem59 42701 stoweidlem60 42702 stoweidlem61 42703 stoweidlem62 42704 stoweid 42705 |
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