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 5747 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
4 | rfcnpre1.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
5 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥(,) | |
6 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥+∞ | |
7 | 4, 5, 6 | nfov 7243 | . . . . 5 ⊢ Ⅎ𝑥(𝐵(,)+∞) |
8 | 3, 7 | nfima 5937 | . . . 4 ⊢ Ⅎ𝑥(◡𝐹 “ (𝐵(,)+∞)) |
9 | nfrab1 3296 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} | |
10 | rfcnpre1.8 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
11 | cntop1 22137 | . . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
12 | 10, 11 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Top) |
13 | rfcnpre1.5 | . . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝐽 | |
14 | istopon 21809 | . . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
15 | 12, 13, 14 | sylanblrc 593 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
16 | rfcnpre1.4 | . . . . . . . . . . . 12 ⊢ 𝐾 = (topGen‘ran (,)) | |
17 | retopon 23661 | . . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
18 | 16, 17 | eqeltri 2834 | . . . . . . . . . . 11 ⊢ 𝐾 ∈ (TopOn‘ℝ) |
19 | iscn 22132 | . . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℝ)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) | |
20 | 15, 18, 19 | sylancl 589 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) |
21 | 10, 20 | mpbid 235 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)) |
22 | 21 | simpld 498 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
23 | 22 | ffvelrnda 6904 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ℝ) |
24 | rfcnpre1.7 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
25 | elioopnf 13031 | . . . . . . . . 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 543 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝑥) ∈ ℝ) → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ 𝐵 < (𝐹‘𝑥))) |
28 | 23, 27 | syldan 594 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ 𝐵 < (𝐹‘𝑥))) |
29 | 28 | pm5.32da 582 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (𝐵(,)+∞)) ↔ (𝑥 ∈ 𝑋 ∧ 𝐵 < (𝐹‘𝑥)))) |
30 | ffn 6545 | . . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → 𝐹 Fn 𝑋) | |
31 | elpreima 6878 | . . . . . 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 3290 | . . . . . 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 314 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (𝐵(,)+∞)) ↔ 𝑥 ∈ {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)})) |
36 | 1, 8, 9, 35 | eqrd 3920 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)}) |
37 | rfcnpre1.6 | . . 3 ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} | |
38 | 36, 37 | eqtr4di 2796 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) = 𝐴) |
39 | iooretop 23663 | . . . 4 ⊢ (𝐵(,)+∞) ∈ (topGen‘ran (,)) | |
40 | 39, 16 | eleqtrri 2837 | . . 3 ⊢ (𝐵(,)+∞) ∈ 𝐾 |
41 | cnima 22162 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐵(,)+∞) ∈ 𝐾) → (◡𝐹 “ (𝐵(,)+∞)) ∈ 𝐽) | |
42 | 10, 40, 41 | sylancl 589 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) ∈ 𝐽) |
43 | 38, 42 | eqeltrrd 2839 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2884 ∀wral 3061 {crab 3065 ∪ cuni 4819 class class class wbr 5053 ◡ccnv 5550 ran crn 5552 “ cima 5554 Fn wfn 6375 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 +∞cpnf 10864 ℝ*cxr 10866 < clt 10867 (,)cioo 12935 topGenctg 16942 Topctop 21790 TopOnctopon 21807 Cn ccn 22121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 df-ioo 12939 df-topgen 16948 df-top 21791 df-topon 21808 df-bases 21843 df-cn 22124 |
This theorem is referenced by: stoweidlem46 43262 |
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