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Mirrors > Home > MPE Home > Th. List > flimclsi | Structured version Visualization version GIF version |
Description: The convergent points of a filter are a subset of the closure of any of the filter sets. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
Ref | Expression |
---|---|
flimclsi | ⊢ (𝑆 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | flimfil 23993 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
3 | 2 | ad2antlr 727 | . . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
4 | flimnei 23991 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦 ∈ 𝐹) | |
5 | 4 | adantll 714 | . . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦 ∈ 𝐹) |
6 | simpll 767 | . . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 ∈ 𝐹) | |
7 | filinn0 23884 | . . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐽) ∧ 𝑦 ∈ 𝐹 ∧ 𝑆 ∈ 𝐹) → (𝑦 ∩ 𝑆) ≠ ∅) | |
8 | 3, 5, 6, 7 | syl3anc 1370 | . . . . 5 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → (𝑦 ∩ 𝑆) ≠ ∅) |
9 | 8 | ralrimiva 3144 | . . . 4 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅) |
10 | flimtop 23989 | . . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) | |
11 | 10 | adantl 481 | . . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝐽 ∈ Top) |
12 | filelss 23876 | . . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐽) ∧ 𝑆 ∈ 𝐹) → 𝑆 ⊆ ∪ 𝐽) | |
13 | 12 | ancoms 458 | . . . . . 6 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) → 𝑆 ⊆ ∪ 𝐽) |
14 | 2, 13 | sylan2 593 | . . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑆 ⊆ ∪ 𝐽) |
15 | 1 | flimelbas 23992 | . . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ∪ 𝐽) |
16 | 15 | adantl 481 | . . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ∪ 𝐽) |
17 | 1 | neindisj2 23147 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅)) |
18 | 11, 14, 16, 17 | syl3anc 1370 | . . . 4 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅)) |
19 | 9, 18 | mpbird 257 | . . 3 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)) |
20 | 19 | ex 412 | . 2 ⊢ (𝑆 ∈ 𝐹 → (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆))) |
21 | 20 | ssrdv 4001 | 1 ⊢ (𝑆 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 {csn 4631 ∪ cuni 4912 ‘cfv 6563 (class class class)co 7431 Topctop 22915 clsccl 23042 neicnei 23121 Filcfil 23869 fLim cflim 23958 |
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 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 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-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-fbas 21379 df-top 22916 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-fil 23870 df-flim 23963 |
This theorem is referenced by: flimcls 24009 flimfcls 24050 cnextcn 24091 cmetss 25364 minveclem4 25480 |
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