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 2921 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
4 | cntop1 21840 | . . . 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 21517 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑇)) |
8 | 5, 7 | sylib 220 | . 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑇)) |
9 | fcnre.1 | . . . 4 ⊢ 𝐾 = (topGen‘ran (,)) | |
10 | retopon 23364 | . . . 4 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
11 | 9, 10 | eqeltri 2907 | . . 3 ⊢ 𝐾 ∈ (TopOn‘ℝ) |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℝ)) |
13 | cnf2 21849 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑇) ∧ 𝐾 ∈ (TopOn‘ℝ) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑇⟶ℝ) | |
14 | 8, 12, 3, 13 | syl3anc 1366 | 1 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ∪ cuni 4830 ran crn 5549 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ℝcr 10528 (,)cioo 12730 topGenctg 16703 Topctop 21493 TopOnctopon 21510 Cn ccn 21824 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-pre-lttri 10603 ax-pre-lttrn 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 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 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-ioo 12734 df-topgen 16709 df-top 21494 df-topon 21511 df-bases 21546 df-cn 21827 |
This theorem is referenced by: rfcnpre2 41279 cncmpmax 41280 rfcnpre3 41281 rfcnpre4 41282 rfcnnnub 41284 stoweidlem28 42304 stoweidlem29 42305 stoweidlem36 42312 stoweidlem43 42319 stoweidlem44 42320 stoweidlem47 42323 stoweidlem52 42328 stoweidlem57 42333 stoweidlem59 42335 stoweidlem60 42336 stoweidlem61 42337 stoweidlem62 42338 stoweid 42339 |
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