| 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 5889 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
| 4 | rfcnpre1.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
| 5 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑥(,) | |
| 6 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑥+∞ | |
| 7 | 4, 5, 6 | nfov 7461 | . . . . 5 ⊢ Ⅎ𝑥(𝐵(,)+∞) |
| 8 | 3, 7 | nfima 6086 | . . . 4 ⊢ Ⅎ𝑥(◡𝐹 “ (𝐵(,)+∞)) |
| 9 | nfrab1 3457 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} | |
| 10 | rfcnpre1.8 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
| 11 | cntop1 23248 | . . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
| 12 | 10, 11 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Top) |
| 13 | rfcnpre1.5 | . . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝐽 | |
| 14 | istopon 22918 | . . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
| 15 | 12, 13, 14 | sylanblrc 590 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
| 16 | rfcnpre1.4 | . . . . . . . . . . . 12 ⊢ 𝐾 = (topGen‘ran (,)) | |
| 17 | retopon 24784 | . . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
| 18 | 16, 17 | eqeltri 2837 | . . . . . . . . . . 11 ⊢ 𝐾 ∈ (TopOn‘ℝ) |
| 19 | iscn 23243 | . . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℝ)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) | |
| 20 | 15, 18, 19 | sylancl 586 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) |
| 21 | 10, 20 | mpbid 232 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹:𝑋⟶ℝ ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)) |
| 22 | 21 | simpld 494 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
| 23 | 22 | ffvelcdmda 7104 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ℝ) |
| 24 | rfcnpre1.7 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
| 25 | elioopnf 13483 | . . . . . . . . 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 539 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝑥) ∈ ℝ) → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ 𝐵 < (𝐹‘𝑥))) |
| 28 | 23, 27 | syldan 591 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) ∈ (𝐵(,)+∞) ↔ 𝐵 < (𝐹‘𝑥))) |
| 29 | 28 | pm5.32da 579 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (𝐵(,)+∞)) ↔ (𝑥 ∈ 𝑋 ∧ 𝐵 < (𝐹‘𝑥)))) |
| 30 | ffn 6736 | . . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → 𝐹 Fn 𝑋) | |
| 31 | elpreima 7078 | . . . . . 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 3458 | . . . . . 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 311 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (𝐵(,)+∞)) ↔ 𝑥 ∈ {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)})) |
| 36 | 1, 8, 9, 35 | eqrd 4003 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)}) |
| 37 | rfcnpre1.6 | . . 3 ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)} | |
| 38 | 36, 37 | eqtr4di 2795 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) = 𝐴) |
| 39 | iooretop 24786 | . . . 4 ⊢ (𝐵(,)+∞) ∈ (topGen‘ran (,)) | |
| 40 | 39, 16 | eleqtrri 2840 | . . 3 ⊢ (𝐵(,)+∞) ∈ 𝐾 |
| 41 | cnima 23273 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐵(,)+∞) ∈ 𝐾) → (◡𝐹 “ (𝐵(,)+∞)) ∈ 𝐽) | |
| 42 | 10, 40, 41 | sylancl 586 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐵(,)+∞)) ∈ 𝐽) |
| 43 | 38, 42 | eqeltrrd 2842 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 {crab 3436 ∪ cuni 4907 class class class wbr 5143 ◡ccnv 5684 ran crn 5686 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 (,)cioo 13387 topGenctg 17482 Topctop 22899 TopOnctopon 22916 Cn ccn 23232 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-ioo 13391 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 df-cn 23235 |
| This theorem is referenced by: stoweidlem46 46061 |
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