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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rfcnpre2 | Structured version Visualization version GIF version |
Description: If 𝐹 is a continuous function with respect to the standard topology, then the preimage A of the values smaller than a given extended real 𝐵, is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
rfcnpre2.1 | ⊢ Ⅎ𝑥𝐵 |
rfcnpre2.2 | ⊢ Ⅎ𝑥𝐹 |
rfcnpre2.3 | ⊢ Ⅎ𝑥𝜑 |
rfcnpre2.4 | ⊢ 𝐾 = (topGen‘ran (,)) |
rfcnpre2.5 | ⊢ 𝑋 = ∪ 𝐽 |
rfcnpre2.6 | ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵} |
rfcnpre2.7 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
rfcnpre2.8 | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
Ref | Expression |
---|---|
rfcnpre2 | ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rfcnpre2.3 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | rfcnpre2.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
3 | 2 | nfcnv 5876 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
4 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑥-∞ | |
5 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑥(,) | |
6 | rfcnpre2.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
7 | 4, 5, 6 | nfov 7435 | . . . . 5 ⊢ Ⅎ𝑥(-∞(,)𝐵) |
8 | 3, 7 | nfima 6065 | . . . 4 ⊢ Ⅎ𝑥(◡𝐹 “ (-∞(,)𝐵)) |
9 | nfrab1 3451 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵} | |
10 | rfcnpre2.4 | . . . . . . . . 9 ⊢ 𝐾 = (topGen‘ran (,)) | |
11 | rfcnpre2.5 | . . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽 | |
12 | eqid 2732 | . . . . . . . . 9 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾) | |
13 | rfcnpre2.8 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
14 | 10, 11, 12, 13 | fcnre 43694 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
15 | 14 | ffvelcdmda 7083 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ℝ) |
16 | rfcnpre2.7 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
17 | elioomnf 13417 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ* → ((𝐹‘𝑥) ∈ (-∞(,)𝐵) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (𝐹‘𝑥) < 𝐵))) | |
18 | 16, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝑥) ∈ (-∞(,)𝐵) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (𝐹‘𝑥) < 𝐵))) |
19 | 18 | baibd 540 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝑥) ∈ ℝ) → ((𝐹‘𝑥) ∈ (-∞(,)𝐵) ↔ (𝐹‘𝑥) < 𝐵)) |
20 | 15, 19 | syldan 591 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) ∈ (-∞(,)𝐵) ↔ (𝐹‘𝑥) < 𝐵)) |
21 | 20 | pm5.32da 579 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (-∞(,)𝐵)) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) < 𝐵))) |
22 | ffn 6714 | . . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → 𝐹 Fn 𝑋) | |
23 | elpreima 7056 | . . . . . 6 ⊢ (𝐹 Fn 𝑋 → (𝑥 ∈ (◡𝐹 “ (-∞(,)𝐵)) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (-∞(,)𝐵)))) | |
24 | 14, 22, 23 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (-∞(,)𝐵)) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (-∞(,)𝐵)))) |
25 | rabid 3452 | . . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵} ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) < 𝐵)) | |
26 | 25 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵} ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) < 𝐵))) |
27 | 21, 24, 26 | 3bitr4d 310 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (-∞(,)𝐵)) ↔ 𝑥 ∈ {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵})) |
28 | 1, 8, 9, 27 | eqrd 4000 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵}) |
29 | rfcnpre2.6 | . . 3 ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵} | |
30 | 28, 29 | eqtr4di 2790 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = 𝐴) |
31 | iooretop 24273 | . . . . 5 ⊢ (-∞(,)𝐵) ∈ (topGen‘ran (,)) | |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → (-∞(,)𝐵) ∈ (topGen‘ran (,))) |
33 | 32, 10 | eleqtrrdi 2844 | . . 3 ⊢ (𝜑 → (-∞(,)𝐵) ∈ 𝐾) |
34 | cnima 22760 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (-∞(,)𝐵) ∈ 𝐾) → (◡𝐹 “ (-∞(,)𝐵)) ∈ 𝐽) | |
35 | 13, 33, 34 | syl2anc 584 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) ∈ 𝐽) |
36 | 30, 35 | eqeltrrd 2834 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 {crab 3432 ∪ cuni 4907 class class class wbr 5147 ◡ccnv 5674 ran crn 5676 “ cima 5678 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℝcr 11105 -∞cmnf 11242 ℝ*cxr 11243 < clt 11244 (,)cioo 13320 topGenctg 17379 Cn ccn 22719 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-ioo 13324 df-topgen 17385 df-top 22387 df-topon 22404 df-bases 22440 df-cn 22722 |
This theorem is referenced by: stoweidlem52 44754 cnfsmf 45442 |
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