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Theorem flimfil 23964
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 23959 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
32simplbi 496 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋))
43simp2d 1140 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ran Fil)
5 filunirn 23877 . . 3 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
64, 5sylib 217 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐹))
73simp3d 1141 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
8 sspwuni 5108 . . . . 5 (𝐹 ⊆ 𝒫 𝑋 𝐹𝑋)
97, 8sylib 217 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹𝑋)
10 flimneiss 23961 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
11 flimtop 23960 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
121topopn 22899 . . . . . . . 8 (𝐽 ∈ Top → 𝑋𝐽)
1311, 12syl 17 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋𝐽)
141flimelbas 23963 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴𝑋)
15 opnneip 23114 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋𝐽𝐴𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
1611, 13, 14, 15syl3anc 1368 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
1710, 16sseldd 3980 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋𝐹)
18 elssuni 4945 . . . . 5 (𝑋𝐹𝑋 𝐹)
1917, 18syl 17 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 𝐹)
209, 19eqssd 3997 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 = 𝑋)
2120fveq2d 6905 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (Fil‘ 𝐹) = (Fil‘𝑋))
226, 21eleqtrd 2828 1 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  wss 3947  𝒫 cpw 4607  {csn 4633   cuni 4913  ran crn 5683  cfv 6554  (class class class)co 7424  Topctop 22886  neicnei 23092  Filcfil 23840   fLim cflim 23929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-fbas 21340  df-top 22887  df-nei 23093  df-fil 23841  df-flim 23934
This theorem is referenced by:  flimtopon  23965  flimss1  23968  flimclsi  23973  hausflimlem  23974  flimsncls  23981  cnpflfi  23994  flimfcls  24021  flimcfil  25333
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