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Theorem flimneiss 22575
Description: A filter contains the neighborhood filter as a subfilter. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
flimneiss (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)

Proof of Theorem flimneiss
StepHypRef Expression
1 eqid 2801 . . . 4 𝐽 = 𝐽
21elflim2 22573 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝐽) ∧ (𝐴 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
32simprbi 500 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐴 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
43simprd 499 1 (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2112  wss 3884  𝒫 cpw 4500  {csn 4528   cuni 4803  ran crn 5524  cfv 6328  (class class class)co 7139  Topctop 21502  neicnei 21706  Filcfil 22454   fLim cflim 22543
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-top 21503  df-flim 22548
This theorem is referenced by:  flimnei  22576  flimfil  22578  flimss2  22581  flimss1  22582  flimcf  22591
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