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 2920 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
4 | cntop1 21776 | . . . 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 21453 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑇)) |
8 | 5, 7 | sylib 219 | . 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑇)) |
9 | fcnre.1 | . . . 4 ⊢ 𝐾 = (topGen‘ran (,)) | |
10 | retopon 23299 | . . . 4 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
11 | 9, 10 | eqeltri 2906 | . . 3 ⊢ 𝐾 ∈ (TopOn‘ℝ) |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℝ)) |
13 | cnf2 21785 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑇) ∧ 𝐾 ∈ (TopOn‘ℝ) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑇⟶ℝ) | |
14 | 8, 12, 3, 13 | syl3anc 1363 | 1 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∪ cuni 4830 ran crn 5549 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 (,)cioo 12726 topGenctg 16699 Topctop 21429 TopOnctopon 21446 Cn ccn 21760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-ioo 12730 df-topgen 16705 df-top 21430 df-topon 21447 df-bases 21482 df-cn 21763 |
This theorem is referenced by: rfcnpre2 41165 cncmpmax 41166 rfcnpre3 41167 rfcnpre4 41168 rfcnnnub 41170 stoweidlem28 42190 stoweidlem29 42191 stoweidlem36 42198 stoweidlem43 42205 stoweidlem44 42206 stoweidlem47 42209 stoweidlem52 42214 stoweidlem57 42219 stoweidlem59 42221 stoweidlem60 42222 stoweidlem61 42223 stoweidlem62 42224 stoweid 42225 |
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