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Theorem flimtopon 21946
 Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
flimtopon (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))

Proof of Theorem flimtopon
StepHypRef Expression
1 flimtop 21941 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 istopon 20890 . . . 4 (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
32baib 982 . . 3 (𝐽 ∈ Top → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = 𝐽))
41, 3syl 17 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = 𝐽))
5 eqid 2748 . . . . 5 𝐽 = 𝐽
65flimfil 21945 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
7 fveq2 6340 . . . . 5 (𝑋 = 𝐽 → (Fil‘𝑋) = (Fil‘ 𝐽))
87eleq2d 2813 . . . 4 (𝑋 = 𝐽 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘ 𝐽)))
96, 8syl5ibrcom 237 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑋 = 𝐽𝐹 ∈ (Fil‘𝑋)))
10 filunibas 21857 . . . . 5 (𝐹 ∈ (Fil‘ 𝐽) → 𝐹 = 𝐽)
116, 10syl 17 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 = 𝐽)
12 filunibas 21857 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
1312eqeq1d 2750 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ( 𝐹 = 𝐽𝑋 = 𝐽))
1411, 13syl5ibcom 235 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐹 ∈ (Fil‘𝑋) → 𝑋 = 𝐽))
159, 14impbid 202 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑋 = 𝐽𝐹 ∈ (Fil‘𝑋)))
164, 15bitrd 268 1 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1620   ∈ wcel 2127  ∪ cuni 4576  ‘cfv 6037  (class class class)co 6801  Topctop 20871  TopOnctopon 20888  Filcfil 21821   fLim cflim 21910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-fbas 19916  df-top 20872  df-topon 20889  df-nei 21075  df-fil 21822  df-flim 21915 This theorem is referenced by:  fclsfnflim  22003
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