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Theorem fclstopon 22658
 Description: Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
fclstopon (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))

Proof of Theorem fclstopon
StepHypRef Expression
1 fclstop 22657 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
2 istopon 21558 . . . 4 (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
32baib 539 . . 3 (𝐽 ∈ Top → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = 𝐽))
41, 3syl 17 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = 𝐽))
5 eqid 2798 . . . . 5 𝐽 = 𝐽
65fclsfil 22656 . . . 4 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
7 fveq2 6655 . . . . 5 (𝑋 = 𝐽 → (Fil‘𝑋) = (Fil‘ 𝐽))
87eleq2d 2875 . . . 4 (𝑋 = 𝐽 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘ 𝐽)))
96, 8syl5ibrcom 250 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑋 = 𝐽𝐹 ∈ (Fil‘𝑋)))
10 filunibas 22527 . . . . 5 (𝐹 ∈ (Fil‘ 𝐽) → 𝐹 = 𝐽)
116, 10syl 17 . . . 4 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 = 𝐽)
12 filunibas 22527 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
1312eqeq1d 2800 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ( 𝐹 = 𝐽𝑋 = 𝐽))
1411, 13syl5ibcom 248 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐹 ∈ (Fil‘𝑋) → 𝑋 = 𝐽))
159, 14impbid 215 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑋 = 𝐽𝐹 ∈ (Fil‘𝑋)))
164, 15bitrd 282 1 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∪ cuni 4804  ‘cfv 6332  (class class class)co 7145  Topctop 21539  TopOnctopon 21556  Filcfil 22491   fClus cfcls 22582 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-int 4843  df-iin 4888  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  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 6291  df-fun 6334  df-fn 6335  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-fbas 20109  df-topon 21557  df-fil 22492  df-fcls 22587 This theorem is referenced by:  fclsopni  22661  fclselbas  22662  fclsss1  22668  fclsss2  22669  fclscf  22671
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