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 5742 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
4 | rfcnpre1.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
5 | nfcv 2976 | . . . . . 6 ⊢ Ⅎ𝑥(,) | |
6 | nfcv 2976 | . . . . . 6 ⊢ Ⅎ𝑥+∞ | |
7 | 4, 5, 6 | nfov 7179 | . . . . 5 ⊢ Ⅎ𝑥(𝐵(,)+∞) |
8 | 3, 7 | nfima 5930 | . . . 4 ⊢ Ⅎ𝑥(◡𝐹 “ (𝐵(,)+∞)) |
9 | nfrab1 3383 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} | |
10 | rfcnpre1.8 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
11 | cntop1 21843 | . . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
12 | 10, 11 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Top) |
13 | rfcnpre1.5 | . . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝐽 | |
14 | istopon 21515 | . . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
15 | 12, 13, 14 | sylanblrc 592 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
16 | rfcnpre1.4 | . . . . . . . . . . . 12 ⊢ 𝐾 = (topGen‘ran (,)) | |
17 | retopon 23367 | . . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
18 | 16, 17 | eqeltri 2908 | . . . . . . . . . . 11 ⊢ 𝐾 ∈ (TopOn‘ℝ) |
19 | iscn 21838 | . . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℝ)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) | |
20 | 15, 18, 19 | sylancl 588 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) |
21 | 10, 20 | mpbid 234 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)) |
22 | 21 | simpld 497 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
23 | 22 | ffvelrnda 6844 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ℝ) |
24 | rfcnpre1.7 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
25 | elioopnf 12825 | . . . . . . . . 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 542 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝑥) ∈ ℝ) → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ 𝐵 < (𝐹‘𝑥))) |
28 | 23, 27 | syldan 593 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ 𝐵 < (𝐹‘𝑥))) |
29 | 28 | pm5.32da 581 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (𝐵(,)+∞)) ↔ (𝑥 ∈ 𝑋 ∧ 𝐵 < (𝐹‘𝑥)))) |
30 | ffn 6507 | . . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → 𝐹 Fn 𝑋) | |
31 | elpreima 6821 | . . . . . 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 3377 | . . . . . 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 313 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (𝐵(,)+∞)) ↔ 𝑥 ∈ {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)})) |
36 | 1, 8, 9, 35 | eqrd 3979 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)}) |
37 | rfcnpre1.6 | . . 3 ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} | |
38 | 36, 37 | syl6eqr 2873 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) = 𝐴) |
39 | iooretop 23369 | . . . 4 ⊢ (𝐵(,)+∞) ∈ (topGen‘ran (,)) | |
40 | 39, 16 | eleqtrri 2911 | . . 3 ⊢ (𝐵(,)+∞) ∈ 𝐾 |
41 | cnima 21868 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐵(,)+∞) ∈ 𝐾) → (◡𝐹 “ (𝐵(,)+∞)) ∈ 𝐽) | |
42 | 10, 40, 41 | sylancl 588 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) ∈ 𝐽) |
43 | 38, 42 | eqeltrrd 2913 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 Ⅎwnfc 2960 ∀wral 3137 {crab 3141 ∪ cuni 4831 class class class wbr 5059 ◡ccnv 5547 ran crn 5549 “ cima 5551 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 ℝcr 10529 +∞cpnf 10665 ℝ*cxr 10667 < clt 10668 (,)cioo 12732 topGenctg 16706 Topctop 21496 TopOnctopon 21513 Cn ccn 21827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 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-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-sup 8899 df-inf 8900 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-ioo 12736 df-topgen 16712 df-top 21497 df-topon 21514 df-bases 21549 df-cn 21830 |
This theorem is referenced by: stoweidlem46 42405 |
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