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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 2736 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
8 | 7 | cldss 22378 | . . . . 5 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽) |
9 | 6, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) |
10 | toponuni 22261 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
12 | 9, 11 | sseqtrrd 3985 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
13 | 1, 2, 3, 4, 5, 12 | lmcls 22651 | . 2 ⊢ (𝜑 → 𝑃 ∈ ((cls‘𝐽)‘𝑆)) |
14 | cldcls 22391 | . . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆) | |
15 | 6, 14 | syl 17 | . 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝑆) = 𝑆) |
16 | 13, 15 | eleqtrd 2840 | 1 ⊢ (𝜑 → 𝑃 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 ∪ cuni 4865 class class class wbr 5105 ‘cfv 6496 ℤcz 12498 ℤ≥cuz 12762 TopOnctopon 22257 Clsdccld 22365 clsccl 22367 ⇝𝑡clm 22575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-pre-lttri 11124 ax-pre-lttrn 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7920 df-2nd 7921 df-er 8647 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-neg 11387 df-z 12499 df-uz 12763 df-top 22241 df-topon 22258 df-cld 22368 df-ntr 22369 df-cls 22370 df-lm 22578 |
This theorem is referenced by: 1stckgen 22903 lmle 24663 |
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