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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 3989 | . . . 4 ⊢ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴}) | |
2 | 1 | jctr 527 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴}))) |
3 | 2 | adantl 484 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴}))) |
4 | simpl 485 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) | |
5 | snssi 4741 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ⊆ 𝑋) | |
6 | 5 | adantl 484 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ⊆ 𝑋) |
7 | snnzg 4710 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ≠ ∅) | |
8 | 7 | adantl 484 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ≠ ∅) |
9 | neifil 22488 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋)) | |
10 | 4, 6, 8, 9 | syl3anc 1367 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋)) |
11 | elflim 22579 | . . 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 593 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴})))) |
13 | 3, 12 | mpbird 259 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 ≠ wne 3016 ⊆ wss 3936 ∅c0 4291 {csn 4567 ‘cfv 6355 (class class class)co 7156 TopOnctopon 21518 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 ax-un 7461 |
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-topon 21519 df-nei 21706 df-fil 22454 df-flim 22547 |
This theorem is referenced by: flimcf 22590 cnpflf2 22608 cnpflf 22609 flfcntr 22651 |
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