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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 22572 | . . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))) |
3 | 2 | simplbi 500 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋)) |
4 | 3 | simp2d 1139 | . . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ ∪ ran Fil) |
5 | filunirn 22490 | . . 3 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹)) | |
6 | 4, 5 | sylib 220 | . 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐹)) |
7 | 3 | simp3d 1140 | . . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ⊆ 𝒫 𝑋) |
8 | sspwuni 5022 | . . . . 5 ⊢ (𝐹 ⊆ 𝒫 𝑋 ↔ ∪ 𝐹 ⊆ 𝑋) | |
9 | 7, 8 | sylib 220 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∪ 𝐹 ⊆ 𝑋) |
10 | flimneiss 22574 | . . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹) | |
11 | flimtop 22573 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) | |
12 | 1 | topopn 21514 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
13 | 11, 12 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ 𝐽) |
14 | 1 | flimelbas 22576 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ 𝑋) |
15 | opnneip 21727 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴})) | |
16 | 11, 13, 14, 15 | syl3anc 1367 | . . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴})) |
17 | 10, 16 | sseldd 3968 | . . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ 𝐹) |
18 | elssuni 4868 | . . . . 5 ⊢ (𝑋 ∈ 𝐹 → 𝑋 ⊆ ∪ 𝐹) | |
19 | 17, 18 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ⊆ ∪ 𝐹) |
20 | 9, 19 | eqssd 3984 | . . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∪ 𝐹 = 𝑋) |
21 | 20 | fveq2d 6674 | . 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (Fil‘∪ 𝐹) = (Fil‘𝑋)) |
22 | 6, 21 | eleqtrd 2915 | 1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 𝒫 cpw 4539 {csn 4567 ∪ cuni 4838 ran crn 5556 ‘cfv 6355 (class class class)co 7156 Topctop 21501 neicnei 21705 Filcfil 22453 fLim cflim 22542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-fbas 20542 df-top 21502 df-nei 21706 df-fil 22454 df-flim 22547 |
This theorem is referenced by: flimtopon 22578 flimss1 22581 flimclsi 22586 hausflimlem 22587 flimsncls 22594 cnpflfi 22607 flimfcls 22634 flimcfil 23917 |
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