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Mirrors > Home > MPE Home > Th. List > lmcls | Structured version Visualization version GIF version |
Description: Any convergent sequence of points in a subset of a topological space converges to a point in the closure of the subset. (Contributed by Mario Carneiro, 30-Dec-2013.) (Revised by Mario Carneiro, 1-May-2014.) |
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) |
lmcls.8 | ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
Ref | Expression |
---|---|
lmcls | ⊢ (𝜑 → 𝑃 ∈ ((cls‘𝐽)‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmcls.5 | . . . . 5 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) | |
2 | lmff.3 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
3 | lmff.1 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | lmff.4 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | 2, 3, 4 | lmbr2 21864 | . . . . 5 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) |
6 | 1, 5 | mpbid 235 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))) |
7 | 6 | simp3d 1141 | . . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
8 | 3 | r19.2uz 14703 | . . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) → ∃𝑘 ∈ 𝑍 (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)) |
9 | lmcls.7 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ 𝑆) | |
10 | inelcm 4372 | . . . . . . . . . 10 ⊢ (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐹‘𝑘) ∈ 𝑆) → (𝑢 ∩ 𝑆) ≠ ∅) | |
11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐹‘𝑘) ∈ 𝑆) → (𝑢 ∩ 𝑆) ≠ ∅)) |
12 | 9, 11 | mpan2d 693 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) ∈ 𝑢 → (𝑢 ∩ 𝑆) ≠ ∅)) |
13 | 12 | adantld 494 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) → (𝑢 ∩ 𝑆) ≠ ∅)) |
14 | 13 | rexlimdva 3243 | . . . . . 6 ⊢ (𝜑 → (∃𝑘 ∈ 𝑍 (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) → (𝑢 ∩ 𝑆) ≠ ∅)) |
15 | 8, 14 | syl5 34 | . . . . 5 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) → (𝑢 ∩ 𝑆) ≠ ∅)) |
16 | 15 | imim2d 57 | . . . 4 ⊢ (𝜑 → ((𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)) → (𝑃 ∈ 𝑢 → (𝑢 ∩ 𝑆) ≠ ∅))) |
17 | 16 | ralimdv 3145 | . . 3 ⊢ (𝜑 → (∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)) → ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → (𝑢 ∩ 𝑆) ≠ ∅))) |
18 | 7, 17 | mpd 15 | . 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → (𝑢 ∩ 𝑆) ≠ ∅)) |
19 | topontop 21518 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
20 | 2, 19 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
21 | lmcls.8 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | |
22 | toponuni 21519 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
23 | 2, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
24 | 21, 23 | sseqtrd 3955 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) |
25 | lmcl 21902 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋) | |
26 | 2, 1, 25 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
27 | 26, 23 | eleqtrd 2892 | . . 3 ⊢ (𝜑 → 𝑃 ∈ ∪ 𝐽) |
28 | eqid 2798 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
29 | 28 | elcls 21678 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ 𝑃 ∈ ∪ 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → (𝑢 ∩ 𝑆) ≠ ∅))) |
30 | 20, 24, 27, 29 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → (𝑢 ∩ 𝑆) ≠ ∅))) |
31 | 18, 30 | mpbird 260 | 1 ⊢ (𝜑 → 𝑃 ∈ ((cls‘𝐽)‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ↑pm cpm 8390 ℂcc 10524 ℤcz 11969 ℤ≥cuz 12231 Topctop 21498 TopOnctopon 21515 clsccl 21623 ⇝𝑡clm 21831 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 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 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-z 11970 df-uz 12232 df-top 21499 df-topon 21516 df-cld 21624 df-ntr 21625 df-cls 21626 df-lm 21834 |
This theorem is referenced by: lmcld 21908 1stcelcls 22066 caublcls 23913 |
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