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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 | syl6eleq 2916 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
4 | cntop1 21415 | . . . 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 21092 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑇)) |
8 | 5, 7 | sylib 210 | . 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑇)) |
9 | fcnre.1 | . . . 4 ⊢ 𝐾 = (topGen‘ran (,)) | |
10 | retopon 22937 | . . . 4 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
11 | 9, 10 | eqeltri 2902 | . . 3 ⊢ 𝐾 ∈ (TopOn‘ℝ) |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℝ)) |
13 | cnf2 21424 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑇) ∧ 𝐾 ∈ (TopOn‘ℝ) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑇⟶ℝ) | |
14 | 8, 12, 3, 13 | syl3anc 1496 | 1 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∪ cuni 4658 ran crn 5343 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ℝcr 10251 (,)cioo 12463 topGenctg 16451 Topctop 21068 TopOnctopon 21085 Cn ccn 21399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-pre-lttri 10326 ax-pre-lttrn 10327 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-ioo 12467 df-topgen 16457 df-top 21069 df-topon 21086 df-bases 21121 df-cn 21402 |
This theorem is referenced by: rfcnpre2 40008 cncmpmax 40009 rfcnpre3 40010 rfcnpre4 40011 rfcnnnub 40013 stoweidlem28 41039 stoweidlem29 41040 stoweidlem36 41047 stoweidlem43 41054 stoweidlem44 41055 stoweidlem47 41058 stoweidlem52 41063 stoweidlem57 41068 stoweidlem59 41070 stoweidlem60 41071 stoweidlem61 41072 stoweidlem62 41073 stoweid 41074 |
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