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

Theorem flimfil 22181
Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flimuni.1 𝑋 = 𝐽
Assertion
Ref Expression
flimfil (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋))

Proof of Theorem flimfil
StepHypRef Expression
1 flimuni.1 . . . . . 6 𝑋 = 𝐽
21elflim2 22176 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
32simplbi 493 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋))
43simp2d 1134 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ran Fil)
5 filunirn 22094 . . 3 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
64, 5sylib 210 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐹))
73simp3d 1135 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
8 sspwuni 4845 . . . . 5 (𝐹 ⊆ 𝒫 𝑋 𝐹𝑋)
97, 8sylib 210 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹𝑋)
10 flimneiss 22178 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
11 flimtop 22177 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
121topopn 21118 . . . . . . . 8 (𝐽 ∈ Top → 𝑋𝐽)
1311, 12syl 17 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋𝐽)
141flimelbas 22180 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴𝑋)
15 opnneip 21331 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋𝐽𝐴𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
1611, 13, 14, 15syl3anc 1439 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
1710, 16sseldd 3821 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋𝐹)
18 elssuni 4702 . . . . 5 (𝑋𝐹𝑋 𝐹)
1917, 18syl 17 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 𝐹)
209, 19eqssd 3837 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 = 𝑋)
2120fveq2d 6450 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (Fil‘ 𝐹) = (Fil‘𝑋))
226, 21eleqtrd 2860 1 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2106  wss 3791  𝒫 cpw 4378  {csn 4397   cuni 4671  ran crn 5356  cfv 6135  (class class class)co 6922  Topctop 21105  neicnei 21309  Filcfil 22057   fLim cflim 22146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-fbas 20139  df-top 21106  df-nei 21310  df-fil 22058  df-flim 22151
This theorem is referenced by:  flimtopon  22182  flimss1  22185  flimclsi  22190  hausflimlem  22191  flimsncls  22198  cnpflfi  22211  flimfcls  22238  flimcfil  23520
  Copyright terms: Public domain W3C validator