MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  flimneiss Structured version   Visualization version   GIF version

Theorem flimneiss 23269
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 2738 . . . 4 𝐽 = 𝐽
21elflim2 23267 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝐽) ∧ (𝐴 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
32simprbi 498 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐴 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
43simprd 497 1 (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wcel 2107  wss 3909  𝒫 cpw 4559  {csn 4585   cuni 4864  ran crn 5633  cfv 6494  (class class class)co 7352  Topctop 22194  neicnei 22400  Filcfil 23148   fLim cflim 23237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6446  df-fun 6496  df-fv 6502  df-ov 7355  df-oprab 7356  df-mpo 7357  df-top 22195  df-flim 23242
This theorem is referenced by:  flimnei  23270  flimfil  23272  flimss2  23275  flimss1  23276  flimcf  23285
  Copyright terms: Public domain W3C validator