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| Mirrors > Home > MPE Home > Th. List > lmcld | Structured version Visualization version GIF version | ||
| Description: Any convergent sequence of points in a closed subset of a topological space converges to a point in the set. (Contributed by Mario Carneiro, 30-Dec-2013.) |
| Ref | Expression |
|---|---|
| lmff.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
| lmff.3 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
| lmff.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
| lmcls.5 | ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) |
| lmcls.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ 𝑆) |
| lmcld.8 | ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) |
| Ref | Expression |
|---|---|
| lmcld | ⊢ (𝜑 → 𝑃 ∈ 𝑆) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lmff.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 2 | lmff.3 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
| 3 | lmff.4 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 4 | lmcls.5 | . . 3 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) | |
| 5 | lmcls.7 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ 𝑆) | |
| 6 | lmcld.8 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) | |
| 7 | eqid 2737 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 8 | 7 | cldss 23037 | . . . . 5 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽) |
| 9 | 6, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) |
| 10 | toponuni 22920 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
| 11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
| 12 | 9, 11 | sseqtrrd 4021 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
| 13 | 1, 2, 3, 4, 5, 12 | lmcls 23310 | . 2 ⊢ (𝜑 → 𝑃 ∈ ((cls‘𝐽)‘𝑆)) |
| 14 | cldcls 23050 | . . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆) | |
| 15 | 6, 14 | syl 17 | . 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝑆) = 𝑆) |
| 16 | 13, 15 | eleqtrd 2843 | 1 ⊢ (𝜑 → 𝑃 ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 ‘cfv 6561 ℤcz 12613 ℤ≥cuz 12878 TopOnctopon 22916 Clsdccld 23024 clsccl 23026 ⇝𝑡clm 23234 |
| 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-pre-lttri 11229 ax-pre-lttrn 11230 |
| 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-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-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-id 5578 df-po 5592 df-so 5593 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-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 df-top 22900 df-topon 22917 df-cld 23027 df-ntr 23028 df-cls 23029 df-lm 23237 |
| This theorem is referenced by: 1stckgen 23562 lmle 25335 |
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