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 5713 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
4 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑥-∞ | |
5 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑥(,) | |
6 | rfcnpre2.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
7 | 4, 5, 6 | nfov 7165 | . . . . 5 ⊢ Ⅎ𝑥(-∞(,)𝐵) |
8 | 3, 7 | nfima 5904 | . . . 4 ⊢ Ⅎ𝑥(◡𝐹 “ (-∞(,)𝐵)) |
9 | nfrab1 3337 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵} | |
10 | rfcnpre2.4 | . . . . . . . . 9 ⊢ 𝐾 = (topGen‘ran (,)) | |
11 | rfcnpre2.5 | . . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽 | |
12 | eqid 2798 | . . . . . . . . 9 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾) | |
13 | rfcnpre2.8 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
14 | 10, 11, 12, 13 | fcnre 41654 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
15 | 14 | ffvelrnda 6828 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ℝ) |
16 | rfcnpre2.7 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
17 | elioomnf 12822 | . . . . . . . . 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 543 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝑥) ∈ ℝ) → ((𝐹‘𝑥) ∈ (-∞(,)𝐵) ↔ (𝐹‘𝑥) < 𝐵)) |
20 | 15, 19 | syldan 594 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) ∈ (-∞(,)𝐵) ↔ (𝐹‘𝑥) < 𝐵)) |
21 | 20 | pm5.32da 582 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (-∞(,)𝐵)) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) < 𝐵))) |
22 | ffn 6487 | . . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → 𝐹 Fn 𝑋) | |
23 | elpreima 6805 | . . . . . 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 3331 | . . . . . 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 314 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (-∞(,)𝐵)) ↔ 𝑥 ∈ {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵})) |
28 | 1, 8, 9, 27 | eqrd 3934 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵}) |
29 | rfcnpre2.6 | . . 3 ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵} | |
30 | 28, 29 | eqtr4di 2851 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = 𝐴) |
31 | iooretop 23371 | . . . . 5 ⊢ (-∞(,)𝐵) ∈ (topGen‘ran (,)) | |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → (-∞(,)𝐵) ∈ (topGen‘ran (,))) |
33 | 32, 10 | eleqtrrdi 2901 | . . 3 ⊢ (𝜑 → (-∞(,)𝐵) ∈ 𝐾) |
34 | cnima 21870 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (-∞(,)𝐵) ∈ 𝐾) → (◡𝐹 “ (-∞(,)𝐵)) ∈ 𝐽) | |
35 | 13, 33, 34 | syl2anc 587 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) ∈ 𝐽) |
36 | 30, 35 | eqeltrrd 2891 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 {crab 3110 ∪ cuni 4800 class class class wbr 5030 ◡ccnv 5518 ran crn 5520 “ cima 5522 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 -∞cmnf 10662 ℝ*cxr 10663 < clt 10664 (,)cioo 12726 topGenctg 16703 Cn ccn 21829 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-ioo 12730 df-topgen 16709 df-top 21499 df-topon 21516 df-bases 21551 df-cn 21832 |
This theorem is referenced by: stoweidlem52 42694 cnfsmf 43374 |
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