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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 2740 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
8 | 7 | cldss 23058 | . . . . 5 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽) |
9 | 6, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) |
10 | toponuni 22941 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
12 | 9, 11 | sseqtrrd 4050 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
13 | 1, 2, 3, 4, 5, 12 | lmcls 23331 | . 2 ⊢ (𝜑 → 𝑃 ∈ ((cls‘𝐽)‘𝑆)) |
14 | cldcls 23071 | . . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆) | |
15 | 6, 14 | syl 17 | . 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝑆) = 𝑆) |
16 | 13, 15 | eleqtrd 2846 | 1 ⊢ (𝜑 → 𝑃 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 ∪ cuni 4931 class class class wbr 5166 ‘cfv 6573 ℤcz 12639 ℤ≥cuz 12903 TopOnctopon 22937 Clsdccld 23045 clsccl 23047 ⇝𝑡clm 23255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-pre-lttri 11258 ax-pre-lttrn 11259 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-er 8763 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-neg 11523 df-z 12640 df-uz 12904 df-top 22921 df-topon 22938 df-cld 23048 df-ntr 23049 df-cls 23050 df-lm 23258 |
This theorem is referenced by: 1stckgen 23583 lmle 25354 |
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