Theorem neiflim 23861

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 3969	. . . 4 ⊢ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴})
2	1	jctr 524	. . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴})))
3	2	adantl 481	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴})))
4		simpl 482	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
5		snssi 4772	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ⊆ 𝑋)
6	5	adantl 481	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ⊆ 𝑋)
7		snnzg 4738	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ≠ ∅)
8	7	adantl 481	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ≠ ∅)
9		neifil 23767	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
10	4, 6, 8, 9	syl3anc 1373	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
11		elflim 23858	. . 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 591	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴}))))
13	3, 12	mpbird 257	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   ≠ wne 2925   ⊆ wss 3914  ∅c0 4296  {csn 4589  ‘cfv 6511  (class class class)co 7387  TopOnctopon 22797  neicnei 22984  Filcfil 23732   fLim cflim 23821

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-fbas 21261  df-top 22781  df-topon 22798  df-nei 22985  df-fil 23733  df-flim 23826

This theorem is referenced by:  flimcf  23869  cnpflf2  23887  cnpflf  23888  flfcntr  23930