Theorem neiflim 23698

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.)

Assertion

Ref	Expression
neiflim	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))

Proof of Theorem neiflim

Step	Hyp	Ref	Expression
1		ssid 4003	. . . 4 ⊢ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴})
2	1	jctr 523	. . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴})))
3	2	adantl 480	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴})))
4		simpl 481	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
5		snssi 4810	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ⊆ 𝑋)
6	5	adantl 480	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ⊆ 𝑋)
7		snnzg 4777	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ≠ ∅)
8	7	adantl 480	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ≠ ∅)
9		neifil 23604	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
10	4, 6, 8, 9	syl3anc 1369	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
11		elflim 23695	. . 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 589	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴}))))
13	3, 12	mpbird 256	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∈ wcel 2104   ≠ wne 2938   ⊆ wss 3947  ∅c0 4321  {csn 4627  ‘cfv 6542  (class class class)co 7411  TopOnctopon 22632  neicnei 22821  Filcfil 23569   fLim cflim 23658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21141  df-top 22616  df-topon 22633  df-nei 22822  df-fil 23570  df-flim 23663

This theorem is referenced by:  flimcf  23706  cnpflf2  23724  cnpflf  23725  flfcntr  23767