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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 2735 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
8 | 7 | cldss 23053 | . . . . 5 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽) |
9 | 6, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) |
10 | toponuni 22936 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
12 | 9, 11 | sseqtrrd 4037 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
13 | 1, 2, 3, 4, 5, 12 | lmcls 23326 | . 2 ⊢ (𝜑 → 𝑃 ∈ ((cls‘𝐽)‘𝑆)) |
14 | cldcls 23066 | . . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆) | |
15 | 6, 14 | syl 17 | . 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝑆) = 𝑆) |
16 | 13, 15 | eleqtrd 2841 | 1 ⊢ (𝜑 → 𝑃 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ∪ cuni 4912 class class class wbr 5148 ‘cfv 6563 ℤcz 12611 ℤ≥cuz 12876 TopOnctopon 22932 Clsdccld 23040 clsccl 23042 ⇝𝑡clm 23250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-neg 11493 df-z 12612 df-uz 12877 df-top 22916 df-topon 22933 df-cld 23043 df-ntr 23044 df-cls 23045 df-lm 23253 |
This theorem is referenced by: 1stckgen 23578 lmle 25349 |
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