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Mirrors > Home > MPE Home > Th. List > flimfil | Structured version Visualization version GIF version |
Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
Ref | Expression |
---|---|
flimuni.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
flimfil | ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flimuni.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | elflim2 22176 | . . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))) |
3 | 2 | simplbi 493 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋)) |
4 | 3 | simp2d 1134 | . . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ ∪ ran Fil) |
5 | filunirn 22094 | . . 3 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹)) | |
6 | 4, 5 | sylib 210 | . 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐹)) |
7 | 3 | simp3d 1135 | . . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ⊆ 𝒫 𝑋) |
8 | sspwuni 4845 | . . . . 5 ⊢ (𝐹 ⊆ 𝒫 𝑋 ↔ ∪ 𝐹 ⊆ 𝑋) | |
9 | 7, 8 | sylib 210 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∪ 𝐹 ⊆ 𝑋) |
10 | flimneiss 22178 | . . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹) | |
11 | flimtop 22177 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) | |
12 | 1 | topopn 21118 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
13 | 11, 12 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ 𝐽) |
14 | 1 | flimelbas 22180 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ 𝑋) |
15 | opnneip 21331 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴})) | |
16 | 11, 13, 14, 15 | syl3anc 1439 | . . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴})) |
17 | 10, 16 | sseldd 3821 | . . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ 𝐹) |
18 | elssuni 4702 | . . . . 5 ⊢ (𝑋 ∈ 𝐹 → 𝑋 ⊆ ∪ 𝐹) | |
19 | 17, 18 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ⊆ ∪ 𝐹) |
20 | 9, 19 | eqssd 3837 | . . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∪ 𝐹 = 𝑋) |
21 | 20 | fveq2d 6450 | . 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (Fil‘∪ 𝐹) = (Fil‘𝑋)) |
22 | 6, 21 | eleqtrd 2860 | 1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ⊆ wss 3791 𝒫 cpw 4378 {csn 4397 ∪ cuni 4671 ran crn 5356 ‘cfv 6135 (class class class)co 6922 Topctop 21105 neicnei 21309 Filcfil 22057 fLim cflim 22146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-fbas 20139 df-top 21106 df-nei 21310 df-fil 22058 df-flim 22151 |
This theorem is referenced by: flimtopon 22182 flimss1 22185 flimclsi 22190 hausflimlem 22191 flimsncls 22198 cnpflfi 22211 flimfcls 22238 flimcfil 23520 |
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