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Theorem flimfil 24235
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 24230 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
32simplbi 502 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋))
43simp2d 1161 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ran Fil)
5 filunirn 24148 . . 3 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
64, 5sylib 221 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐹))
73simp3d 1162 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
8 sspwuni 5060 . . . . 5 (𝐹 ⊆ 𝒫 𝑋 𝐹𝑋)
97, 8sylib 221 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹𝑋)
10 flimneiss 24232 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
11 flimtop 24231 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
121topopn 23171 . . . . . . . 8 (𝐽 ∈ Top → 𝑋𝐽)
1311, 12syl 18 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋𝐽)
141flimelbas 24234 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴𝑋)
15 opnneip 23384 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋𝐽𝐴𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
1611, 13, 14, 15syl3anc 1398 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
1710, 16sseldd 3932 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋𝐹)
18 elssuni 4899 . . . . 5 (𝑋𝐹𝑋 𝐹)
1917, 18syl 18 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 𝐹)
209, 19eqssd 3948 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 = 𝑋)
2120fveq2d 6878 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (Fil‘ 𝐹) = (Fil‘𝑋))
226, 21eleqtrd 2862 1 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wa 401  w3a 1103   = wceq 1570  wcel 2145  wss 3899  𝒫 cpw 4557  {csn 4584   cuni 4867  ran crn 5649  cfv 6528  (class class class)co 7409  Topctop 23158  neicnei 23362  Filcfil 24111   fLim cflim 24200
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-fbas 21622  df-top 23159  df-nei 23363  df-fil 24112  df-flim 24205
This theorem is used by:  flimtopon  24236  flimss1  24239  flimclsi  24244  hausflimlem  24245  flimsncls  24252  cnpflfi  24265  flimfcls  24292  flimcfil  25582
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