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Mirrors > Home > MPE Home > Th. List > neiflim | Structured version Visualization version GIF version |
Description: A point is a limit point of its neighborhood filter. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
Ref | Expression |
---|---|
neiflim | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3937 | . . . 4 ⊢ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴}) | |
2 | 1 | jctr 528 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴}))) |
3 | 2 | adantl 485 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴}))) |
4 | simpl 486 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) | |
5 | snssi 4701 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ⊆ 𝑋) | |
6 | 5 | adantl 485 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ⊆ 𝑋) |
7 | snnzg 4670 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ≠ ∅) | |
8 | 7 | adantl 485 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ≠ ∅) |
9 | neifil 22485 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋)) | |
10 | 4, 6, 8, 9 | syl3anc 1368 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋)) |
11 | elflim 22576 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴})))) | |
12 | 10, 11 | syldan 594 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴})))) |
13 | 3, 12 | mpbird 260 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 {csn 4525 ‘cfv 6324 (class class class)co 7135 TopOnctopon 21515 neicnei 21702 Filcfil 22450 fLim cflim 22539 |
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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 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-fbas 20088 df-top 21499 df-topon 21516 df-nei 21703 df-fil 22451 df-flim 22544 |
This theorem is referenced by: flimcf 22587 cnpflf2 22605 cnpflf 22606 flfcntr 22648 |
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