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Theorem flimneiss 21990
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 2771 . . . 4 𝐽 = 𝐽
21elflim2 21988 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝐽) ∧ (𝐴 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
32simprbi 484 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐴 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
43simprd 483 1 (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071  wcel 2145  wss 3723  𝒫 cpw 4298  {csn 4317   cuni 4575  ran crn 5251  cfv 6030  (class class class)co 6796  Topctop 20918  neicnei 21122  Filcfil 21869   fLim cflim 21958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-top 20919  df-flim 21963
This theorem is referenced by:  flimnei  21991  flimfil  21993  flimss2  21996  flimss1  21997  flimcf  22006
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