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Theorem flimfil 23978
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 23973 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
32simplbi 497 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋))
43simp2d 1143 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ran Fil)
5 filunirn 23891 . . 3 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
64, 5sylib 218 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐹))
73simp3d 1144 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
8 sspwuni 5099 . . . . 5 (𝐹 ⊆ 𝒫 𝑋 𝐹𝑋)
97, 8sylib 218 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹𝑋)
10 flimneiss 23975 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
11 flimtop 23974 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
121topopn 22913 . . . . . . . 8 (𝐽 ∈ Top → 𝑋𝐽)
1311, 12syl 17 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋𝐽)
141flimelbas 23977 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴𝑋)
15 opnneip 23128 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋𝐽𝐴𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
1611, 13, 14, 15syl3anc 1372 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
1710, 16sseldd 3983 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋𝐹)
18 elssuni 4936 . . . . 5 (𝑋𝐹𝑋 𝐹)
1917, 18syl 17 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 𝐹)
209, 19eqssd 4000 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 = 𝑋)
2120fveq2d 6909 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (Fil‘ 𝐹) = (Fil‘𝑋))
226, 21eleqtrd 2842 1 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wss 3950  𝒫 cpw 4599  {csn 4625   cuni 4906  ran crn 5685  cfv 6560  (class class class)co 7432  Topctop 22900  neicnei 23106  Filcfil 23854   fLim cflim 23943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-fbas 21362  df-top 22901  df-nei 23107  df-fil 23855  df-flim 23948
This theorem is referenced by:  flimtopon  23979  flimss1  23982  flimclsi  23987  hausflimlem  23988  flimsncls  23995  cnpflfi  24008  flimfcls  24035  flimcfil  25349
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