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Theorem flimfil 23693
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 = βˆͺ 𝐽
Assertion
Ref Expression
flimfil (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 ∈ (Filβ€˜π‘‹))

Proof of Theorem flimfil
StepHypRef Expression
1 flimuni.1 . . . . . 6 𝑋 = βˆͺ 𝐽
21elflim2 23688 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil ∧ 𝐹 βŠ† 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
32simplbi 498 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ (𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil ∧ 𝐹 βŠ† 𝒫 𝑋))
43simp2d 1143 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 ∈ βˆͺ ran Fil)
5 filunirn 23606 . . 3 (𝐹 ∈ βˆͺ ran Fil ↔ 𝐹 ∈ (Filβ€˜βˆͺ 𝐹))
64, 5sylib 217 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐹))
73simp3d 1144 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 βŠ† 𝒫 𝑋)
8 sspwuni 5103 . . . . 5 (𝐹 βŠ† 𝒫 𝑋 ↔ βˆͺ 𝐹 βŠ† 𝑋)
97, 8sylib 217 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ βˆͺ 𝐹 βŠ† 𝑋)
10 flimneiss 23690 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)
11 flimtop 23689 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐽 ∈ Top)
121topopn 22628 . . . . . . . 8 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
1311, 12syl 17 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝑋 ∈ 𝐽)
141flimelbas 23692 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐴 ∈ 𝑋)
15 opnneip 22843 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) β†’ 𝑋 ∈ ((neiβ€˜π½)β€˜{𝐴}))
1611, 13, 14, 15syl3anc 1371 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝑋 ∈ ((neiβ€˜π½)β€˜{𝐴}))
1710, 16sseldd 3983 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝑋 ∈ 𝐹)
18 elssuni 4941 . . . . 5 (𝑋 ∈ 𝐹 β†’ 𝑋 βŠ† βˆͺ 𝐹)
1917, 18syl 17 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝑋 βŠ† βˆͺ 𝐹)
209, 19eqssd 3999 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ βˆͺ 𝐹 = 𝑋)
2120fveq2d 6895 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ (Filβ€˜βˆͺ 𝐹) = (Filβ€˜π‘‹))
226, 21eleqtrd 2835 1 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908  ran crn 5677  β€˜cfv 6543  (class class class)co 7411  Topctop 22615  neicnei 22821  Filcfil 23569   fLim cflim 23658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21141  df-top 22616  df-nei 22822  df-fil 23570  df-flim 23663
This theorem is referenced by:  flimtopon  23694  flimss1  23697  flimclsi  23702  hausflimlem  23703  flimsncls  23710  cnpflfi  23723  flimfcls  23750  flimcfil  25055
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