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Theorem neiflim 23922
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) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))

Proof of Theorem neiflim
StepHypRef Expression
1 ssid 3999 . . . 4 ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴})
21jctr 523 . . 3 (𝐴𝑋 → (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴})))
32adantl 480 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴})))
4 simpl 481 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
5 snssi 4813 . . . . 5 (𝐴𝑋 → {𝐴} ⊆ 𝑋)
65adantl 480 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → {𝐴} ⊆ 𝑋)
7 snnzg 4780 . . . . 5 (𝐴𝑋 → {𝐴} ≠ ∅)
87adantl 480 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → {𝐴} ≠ ∅)
9 neifil 23828 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
104, 6, 8, 9syl3anc 1368 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
11 elflim 23919 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴}))))
1210, 11syldan 589 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((nei‘𝐽)‘{𝐴}))))
133, 12mpbird 256 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098  wne 2929  wss 3944  c0 4322  {csn 4630  cfv 6549  (class class class)co 7419  TopOnctopon 22856  neicnei 23045  Filcfil 23793   fLim cflim 23882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-fbas 21293  df-top 22840  df-topon 22857  df-nei 23046  df-fil 23794  df-flim 23887
This theorem is referenced by:  flimcf  23930  cnpflf2  23948  cnpflf  23949  flfcntr  23991
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