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Theorem flimfil 22577
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 22572 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
32simplbi 501 . . . 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 22490 . . 3 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
64, 5sylib 221 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐹))
73simp3d 1141 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
8 sspwuni 4988 . . . . 5 (𝐹 ⊆ 𝒫 𝑋 𝐹𝑋)
97, 8sylib 221 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹𝑋)
10 flimneiss 22574 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
11 flimtop 22573 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
121topopn 21514 . . . . . . . 8 (𝐽 ∈ Top → 𝑋𝐽)
1311, 12syl 17 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋𝐽)
141flimelbas 22576 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴𝑋)
15 opnneip 21727 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋𝐽𝐴𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
1611, 13, 14, 15syl3anc 1368 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
1710, 16sseldd 3919 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋𝐹)
18 elssuni 4833 . . . . 5 (𝑋𝐹𝑋 𝐹)
1917, 18syl 17 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 𝐹)
209, 19eqssd 3935 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 = 𝑋)
2120fveq2d 6653 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (Fil‘ 𝐹) = (Fil‘𝑋))
226, 21eleqtrd 2895 1 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  wss 3884  𝒫 cpw 4500  {csn 4528   cuni 4803  ran crn 5524  cfv 6328  (class class class)co 7139  Topctop 21501  neicnei 21705  Filcfil 22453   fLim cflim 22542
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-fbas 20091  df-top 21502  df-nei 21706  df-fil 22454  df-flim 22547
This theorem is referenced by:  flimtopon  22578  flimss1  22581  flimclsi  22586  hausflimlem  22587  flimsncls  22594  cnpflfi  22607  flimfcls  22634  flimcfil  23921
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