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| Mirrors > Home > MPE Home > Th. List > Mathboxes > seppcld | Structured version Visualization version GIF version | ||
| Description: If two sets are precisely separated by a continuous function, then they are closed. An alternate proof involves II ∈ Fre. (Contributed by Zhi Wang, 9-Sep-2024.) |
| Ref | Expression |
|---|---|
| seppsepf.1 | ⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) |
| Ref | Expression |
|---|---|
| seppcld | ⊢ (𝜑 → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | seppsepf.1 | . 2 ⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) | |
| 2 | simprl 771 | . . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → 𝑆 = (◡𝑓 “ {0})) | |
| 3 | simpl 482 | . . . . . 6 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → 𝑓 ∈ (𝐽 Cn II)) | |
| 4 | 0xr 11308 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
| 5 | iccid 13432 | . . . . . . . 8 ⊢ (0 ∈ ℝ* → (0[,]0) = {0}) | |
| 6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ (0[,]0) = {0} |
| 7 | 0le0 12367 | . . . . . . . 8 ⊢ 0 ≤ 0 | |
| 8 | 0le1 11786 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
| 9 | icccldii 48816 | . . . . . . . 8 ⊢ ((0 ≤ 0 ∧ 0 ≤ 1) → (0[,]0) ∈ (Clsd‘II)) | |
| 10 | 7, 8, 9 | mp2an 692 | . . . . . . 7 ⊢ (0[,]0) ∈ (Clsd‘II) |
| 11 | 6, 10 | eqeltrri 2838 | . . . . . 6 ⊢ {0} ∈ (Clsd‘II) |
| 12 | cnclima 23276 | . . . . . 6 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ {0} ∈ (Clsd‘II)) → (◡𝑓 “ {0}) ∈ (Clsd‘𝐽)) | |
| 13 | 3, 11, 12 | sylancl 586 | . . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → (◡𝑓 “ {0}) ∈ (Clsd‘𝐽)) |
| 14 | 2, 13 | eqeltrd 2841 | . . . 4 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → 𝑆 ∈ (Clsd‘𝐽)) |
| 15 | simprr 773 | . . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → 𝑇 = (◡𝑓 “ {1})) | |
| 16 | 1xr 11320 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
| 17 | iccid 13432 | . . . . . . . 8 ⊢ (1 ∈ ℝ* → (1[,]1) = {1}) | |
| 18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (1[,]1) = {1} |
| 19 | 1le1 11891 | . . . . . . . 8 ⊢ 1 ≤ 1 | |
| 20 | icccldii 48816 | . . . . . . . 8 ⊢ ((0 ≤ 1 ∧ 1 ≤ 1) → (1[,]1) ∈ (Clsd‘II)) | |
| 21 | 8, 19, 20 | mp2an 692 | . . . . . . 7 ⊢ (1[,]1) ∈ (Clsd‘II) |
| 22 | 18, 21 | eqeltrri 2838 | . . . . . 6 ⊢ {1} ∈ (Clsd‘II) |
| 23 | cnclima 23276 | . . . . . 6 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ {1} ∈ (Clsd‘II)) → (◡𝑓 “ {1}) ∈ (Clsd‘𝐽)) | |
| 24 | 3, 22, 23 | sylancl 586 | . . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → (◡𝑓 “ {1}) ∈ (Clsd‘𝐽)) |
| 25 | 15, 24 | eqeltrd 2841 | . . . 4 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → 𝑇 ∈ (Clsd‘𝐽)) |
| 26 | 14, 25 | jca 511 | . . 3 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽))) |
| 27 | 26 | rexlimiva 3147 | . 2 ⊢ (∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})) → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽))) |
| 28 | 1, 27 | syl 17 | 1 ⊢ (𝜑 → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {csn 4626 class class class wbr 5143 ◡ccnv 5684 “ cima 5688 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 ℝ*cxr 11294 ≤ cle 11296 [,]cicc 13390 Clsdccld 23024 Cn ccn 23232 IIcii 24901 |
| 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-rep 5279 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-int 4947 df-iun 4993 df-iin 4994 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-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 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-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-rest 17467 df-topgen 17488 df-ordt 17546 df-ps 18611 df-tsr 18612 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cld 23027 df-cn 23235 df-ii 24903 |
| This theorem is referenced by: (None) |
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