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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 2839 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
4 | cntop1 23137 | . . . 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 22812 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑇)) |
8 | 5, 7 | sylib 217 | . 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑇)) |
9 | fcnre.1 | . . . 4 ⊢ 𝐾 = (topGen‘ran (,)) | |
10 | retopon 24673 | . . . 4 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
11 | 9, 10 | eqeltri 2825 | . . 3 ⊢ 𝐾 ∈ (TopOn‘ℝ) |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℝ)) |
13 | cnf2 23146 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑇) ∧ 𝐾 ∈ (TopOn‘ℝ) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑇⟶ℝ) | |
14 | 8, 12, 3, 13 | syl3anc 1369 | 1 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∪ cuni 4903 ran crn 5673 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ℝcr 11131 (,)cioo 13350 topGenctg 17412 Topctop 22788 TopOnctopon 22805 Cn ccn 23121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-pre-lttri 11206 ax-pre-lttrn 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-ioo 13354 df-topgen 17418 df-top 22789 df-topon 22806 df-bases 22842 df-cn 23124 |
This theorem is referenced by: rfcnpre2 44387 cncmpmax 44388 rfcnpre3 44389 rfcnpre4 44390 rfcnnnub 44392 stoweidlem28 45410 stoweidlem29 45411 stoweidlem36 45418 stoweidlem43 45425 stoweidlem44 45426 stoweidlem47 45429 stoweidlem52 45434 stoweidlem57 45439 stoweidlem59 45441 stoweidlem60 45442 stoweidlem61 45443 stoweidlem62 45444 stoweid 45445 |
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