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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rfcnpre1 | 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 greater than a given extended real B is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
rfcnpre1.1 | ⊢ Ⅎ𝑥𝐵 |
rfcnpre1.2 | ⊢ Ⅎ𝑥𝐹 |
rfcnpre1.3 | ⊢ Ⅎ𝑥𝜑 |
rfcnpre1.4 | ⊢ 𝐾 = (topGen‘ran (,)) |
rfcnpre1.5 | ⊢ 𝑋 = ∪ 𝐽 |
rfcnpre1.6 | ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} |
rfcnpre1.7 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
rfcnpre1.8 | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
Ref | Expression |
---|---|
rfcnpre1 | ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rfcnpre1.3 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | rfcnpre1.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
3 | 2 | nfcnv 5892 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
4 | rfcnpre1.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
5 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑥(,) | |
6 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑥+∞ | |
7 | 4, 5, 6 | nfov 7461 | . . . . 5 ⊢ Ⅎ𝑥(𝐵(,)+∞) |
8 | 3, 7 | nfima 6088 | . . . 4 ⊢ Ⅎ𝑥(◡𝐹 “ (𝐵(,)+∞)) |
9 | nfrab1 3454 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} | |
10 | rfcnpre1.8 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
11 | cntop1 23264 | . . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
12 | 10, 11 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Top) |
13 | rfcnpre1.5 | . . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝐽 | |
14 | istopon 22934 | . . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
15 | 12, 13, 14 | sylanblrc 590 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
16 | rfcnpre1.4 | . . . . . . . . . . . 12 ⊢ 𝐾 = (topGen‘ran (,)) | |
17 | retopon 24800 | . . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
18 | 16, 17 | eqeltri 2835 | . . . . . . . . . . 11 ⊢ 𝐾 ∈ (TopOn‘ℝ) |
19 | iscn 23259 | . . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℝ)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) | |
20 | 15, 18, 19 | sylancl 586 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) |
21 | 10, 20 | mpbid 232 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)) |
22 | 21 | simpld 494 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
23 | 22 | ffvelcdmda 7104 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ℝ) |
24 | rfcnpre1.7 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
25 | elioopnf 13480 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ* → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 𝐵 < (𝐹‘𝑥)))) | |
26 | 24, 25 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 𝐵 < (𝐹‘𝑥)))) |
27 | 26 | baibd 539 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝑥) ∈ ℝ) → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ 𝐵 < (𝐹‘𝑥))) |
28 | 23, 27 | syldan 591 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ 𝐵 < (𝐹‘𝑥))) |
29 | 28 | pm5.32da 579 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (𝐵(,)+∞)) ↔ (𝑥 ∈ 𝑋 ∧ 𝐵 < (𝐹‘𝑥)))) |
30 | ffn 6737 | . . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → 𝐹 Fn 𝑋) | |
31 | elpreima 7078 | . . . . . 6 ⊢ (𝐹 Fn 𝑋 → (𝑥 ∈ (◡𝐹 “ (𝐵(,)+∞)) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (𝐵(,)+∞)))) | |
32 | 22, 30, 31 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (𝐵(,)+∞)) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (𝐵(,)+∞)))) |
33 | rabid 3455 | . . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} ↔ (𝑥 ∈ 𝑋 ∧ 𝐵 < (𝐹‘𝑥))) | |
34 | 33 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} ↔ (𝑥 ∈ 𝑋 ∧ 𝐵 < (𝐹‘𝑥)))) |
35 | 29, 32, 34 | 3bitr4d 311 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (𝐵(,)+∞)) ↔ 𝑥 ∈ {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)})) |
36 | 1, 8, 9, 35 | eqrd 4015 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)}) |
37 | rfcnpre1.6 | . . 3 ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} | |
38 | 36, 37 | eqtr4di 2793 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) = 𝐴) |
39 | iooretop 24802 | . . . 4 ⊢ (𝐵(,)+∞) ∈ (topGen‘ran (,)) | |
40 | 39, 16 | eleqtrri 2838 | . . 3 ⊢ (𝐵(,)+∞) ∈ 𝐾 |
41 | cnima 23289 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐵(,)+∞) ∈ 𝐾) → (◡𝐹 “ (𝐵(,)+∞)) ∈ 𝐽) | |
42 | 10, 40, 41 | sylancl 586 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) ∈ 𝐽) |
43 | 38, 42 | eqeltrrd 2840 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ∀wral 3059 {crab 3433 ∪ cuni 4912 class class class wbr 5148 ◡ccnv 5688 ran crn 5690 “ cima 5692 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 +∞cpnf 11290 ℝ*cxr 11292 < clt 11293 (,)cioo 13384 topGenctg 17484 Topctop 22915 TopOnctopon 22932 Cn ccn 23248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-ioo 13388 df-topgen 17490 df-top 22916 df-topon 22933 df-bases 22969 df-cn 23251 |
This theorem is referenced by: stoweidlem46 46002 |
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