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Theorem flimfil 23473
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 23468 . . . . 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 499 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ (𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil ∧ 𝐹 βŠ† 𝒫 𝑋))
43simp2d 1144 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 ∈ βˆͺ ran Fil)
5 filunirn 23386 . . 3 (𝐹 ∈ βˆͺ ran Fil ↔ 𝐹 ∈ (Filβ€˜βˆͺ 𝐹))
64, 5sylib 217 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐹))
73simp3d 1145 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 βŠ† 𝒫 𝑋)
8 sspwuni 5104 . . . . 5 (𝐹 βŠ† 𝒫 𝑋 ↔ βˆͺ 𝐹 βŠ† 𝑋)
97, 8sylib 217 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ βˆͺ 𝐹 βŠ† 𝑋)
10 flimneiss 23470 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)
11 flimtop 23469 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐽 ∈ Top)
121topopn 22408 . . . . . . . 8 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
1311, 12syl 17 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝑋 ∈ 𝐽)
141flimelbas 23472 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐴 ∈ 𝑋)
15 opnneip 22623 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) β†’ 𝑋 ∈ ((neiβ€˜π½)β€˜{𝐴}))
1611, 13, 14, 15syl3anc 1372 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝑋 ∈ ((neiβ€˜π½)β€˜{𝐴}))
1710, 16sseldd 3984 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝑋 ∈ 𝐹)
18 elssuni 4942 . . . . 5 (𝑋 ∈ 𝐹 β†’ 𝑋 βŠ† βˆͺ 𝐹)
1917, 18syl 17 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝑋 βŠ† βˆͺ 𝐹)
209, 19eqssd 4000 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ βˆͺ 𝐹 = 𝑋)
2120fveq2d 6896 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ (Filβ€˜βˆͺ 𝐹) = (Filβ€˜π‘‹))
226, 21eleqtrd 2836 1 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  π’« cpw 4603  {csn 4629  βˆͺ cuni 4909  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  Topctop 22395  neicnei 22601  Filcfil 23349   fLim cflim 23438
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-fbas 20941  df-top 22396  df-nei 22602  df-fil 23350  df-flim 23443
This theorem is referenced by:  flimtopon  23474  flimss1  23477  flimclsi  23482  hausflimlem  23483  flimsncls  23490  cnpflfi  23503  flimfcls  23530  flimcfil  24831
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