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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 2738 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
8 | 7 | cldss 21773 | . . . . 5 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽) |
9 | 6, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) |
10 | toponuni 21658 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
12 | 9, 11 | sseqtrrd 3916 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
13 | 1, 2, 3, 4, 5, 12 | lmcls 22046 | . 2 ⊢ (𝜑 → 𝑃 ∈ ((cls‘𝐽)‘𝑆)) |
14 | cldcls 21786 | . . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆) | |
15 | 6, 14 | syl 17 | . 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝑆) = 𝑆) |
16 | 13, 15 | eleqtrd 2835 | 1 ⊢ (𝜑 → 𝑃 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ⊆ wss 3841 ∪ cuni 4793 class class class wbr 5027 ‘cfv 6333 ℤcz 12055 ℤ≥cuz 12317 TopOnctopon 21654 Clsdccld 21760 clsccl 21762 ⇝𝑡clm 21970 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-pre-lttri 10682 ax-pre-lttrn 10683 |
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 2074 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-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-po 5438 df-so 5439 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-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-1st 7707 df-2nd 7708 df-er 8313 df-pm 8433 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-neg 10944 df-z 12056 df-uz 12318 df-top 21638 df-topon 21655 df-cld 21763 df-ntr 21764 df-cls 21765 df-lm 21973 |
This theorem is referenced by: 1stckgen 22298 lmle 24046 |
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