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 5223 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
4 | rfcnpre1.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
5 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑥(,) | |
6 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑥+∞ | |
7 | 4, 5, 6 | nfov 6575 | . . . . 5 ⊢ Ⅎ𝑥(𝐵(,)+∞) |
8 | 3, 7 | nfima 5393 | . . . 4 ⊢ Ⅎ𝑥(◡𝐹 “ (𝐵(,)+∞)) |
9 | nfrab1 3099 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} | |
10 | rfcnpre1.8 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
11 | cntop1 20854 | . . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
12 | 10, 11 | syl 17 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Top) |
13 | rfcnpre1.5 | . . . . . . . . . . . . 13 ⊢ 𝑋 = ∪ 𝐽 | |
14 | 12, 13 | jctir 559 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) |
15 | istopon 20540 | . . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
16 | 14, 15 | sylibr 223 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
17 | rfcnpre1.4 | . . . . . . . . . . . 12 ⊢ 𝐾 = (topGen‘ran (,)) | |
18 | retopon 22377 | . . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
19 | 17, 18 | eqeltri 2684 | . . . . . . . . . . 11 ⊢ 𝐾 ∈ (TopOn‘ℝ) |
20 | iscn 20849 | . . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℝ)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) | |
21 | 16, 19, 20 | sylancl 693 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) |
22 | 10, 21 | mpbid 221 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)) |
23 | 22 | simpld 474 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
24 | 23 | fnvinran 38196 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ℝ) |
25 | rfcnpre1.7 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
26 | elioopnf 12138 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ* → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 𝐵 < (𝐹‘𝑥)))) | |
27 | 25, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 𝐵 < (𝐹‘𝑥)))) |
28 | 27 | baibd 946 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝑥) ∈ ℝ) → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ 𝐵 < (𝐹‘𝑥))) |
29 | 24, 28 | syldan 486 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ 𝐵 < (𝐹‘𝑥))) |
30 | 29 | pm5.32da 671 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (𝐵(,)+∞)) ↔ (𝑥 ∈ 𝑋 ∧ 𝐵 < (𝐹‘𝑥)))) |
31 | ffn 5958 | . . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → 𝐹 Fn 𝑋) | |
32 | elpreima 6245 | . . . . . 6 ⊢ (𝐹 Fn 𝑋 → (𝑥 ∈ (◡𝐹 “ (𝐵(,)+∞)) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (𝐵(,)+∞)))) | |
33 | 23, 31, 32 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (𝐵(,)+∞)) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (𝐵(,)+∞)))) |
34 | rabid 3095 | . . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} ↔ (𝑥 ∈ 𝑋 ∧ 𝐵 < (𝐹‘𝑥))) | |
35 | 34 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} ↔ (𝑥 ∈ 𝑋 ∧ 𝐵 < (𝐹‘𝑥)))) |
36 | 30, 33, 35 | 3bitr4d 299 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (𝐵(,)+∞)) ↔ 𝑥 ∈ {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)})) |
37 | 1, 8, 9, 36 | eqrd 3586 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)}) |
38 | rfcnpre1.6 | . . 3 ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} | |
39 | 37, 38 | syl6eqr 2662 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) = 𝐴) |
40 | iooretop 22379 | . . . 4 ⊢ (𝐵(,)+∞) ∈ (topGen‘ran (,)) | |
41 | 40, 17 | eleqtrri 2687 | . . 3 ⊢ (𝐵(,)+∞) ∈ 𝐾 |
42 | cnima 20879 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐵(,)+∞) ∈ 𝐾) → (◡𝐹 “ (𝐵(,)+∞)) ∈ 𝐽) | |
43 | 10, 41, 42 | sylancl 693 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) ∈ 𝐽) |
44 | 39, 43 | eqeltrrd 2689 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 ∀wral 2896 {crab 2900 ∪ cuni 4372 class class class wbr 4583 ◡ccnv 5037 ran crn 5039 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 (,)cioo 12046 topGenctg 15921 Topctop 20517 TopOnctopon 20518 Cn ccn 20838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-ioo 12050 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-cn 20841 |
This theorem is referenced by: stoweidlem46 38939 |
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