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 5721 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
4 | nfcv 2899 | . . . . . 6 ⊢ Ⅎ𝑥-∞ | |
5 | nfcv 2899 | . . . . . 6 ⊢ Ⅎ𝑥(,) | |
6 | rfcnpre2.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
7 | 4, 5, 6 | nfov 7200 | . . . . 5 ⊢ Ⅎ𝑥(-∞(,)𝐵) |
8 | 3, 7 | nfima 5911 | . . . 4 ⊢ Ⅎ𝑥(◡𝐹 “ (-∞(,)𝐵)) |
9 | nfrab1 3287 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵} | |
10 | rfcnpre2.4 | . . . . . . . . 9 ⊢ 𝐾 = (topGen‘ran (,)) | |
11 | rfcnpre2.5 | . . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽 | |
12 | eqid 2738 | . . . . . . . . 9 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾) | |
13 | rfcnpre2.8 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
14 | 10, 11, 12, 13 | fcnre 42106 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
15 | 14 | ffvelrnda 6861 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ℝ) |
16 | rfcnpre2.7 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
17 | elioomnf 12918 | . . . . . . . . 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 6504 | . . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → 𝐹 Fn 𝑋) | |
23 | elpreima 6835 | . . . . . 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 3281 | . . . . . 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 3896 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵}) |
29 | rfcnpre2.6 | . . 3 ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵} | |
30 | 28, 29 | eqtr4di 2791 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = 𝐴) |
31 | iooretop 23518 | . . . . 5 ⊢ (-∞(,)𝐵) ∈ (topGen‘ran (,)) | |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → (-∞(,)𝐵) ∈ (topGen‘ran (,))) |
33 | 32, 10 | eleqtrrdi 2844 | . . 3 ⊢ (𝜑 → (-∞(,)𝐵) ∈ 𝐾) |
34 | cnima 22016 | . . 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 2834 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 Ⅎwnf 1790 ∈ wcel 2114 Ⅎwnfc 2879 {crab 3057 ∪ cuni 4796 class class class wbr 5030 ◡ccnv 5524 ran crn 5526 “ cima 5528 Fn wfn 6334 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ℝcr 10614 -∞cmnf 10751 ℝ*cxr 10752 < clt 10753 (,)cioo 12821 topGenctg 16814 Cn ccn 21975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-inf 8980 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-n0 11977 df-z 12063 df-uz 12325 df-q 12431 df-ioo 12825 df-topgen 16820 df-top 21645 df-topon 21662 df-bases 21697 df-cn 21978 |
This theorem is referenced by: stoweidlem52 43135 cnfsmf 43815 |
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