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Theorem fclstop 24019
Description: Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclstop (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)

Proof of Theorem fclstop
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 𝐽 = 𝐽
21isfcls 24017 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
32simp1bi 1146 1 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wral 3061   cuni 4907  cfv 6561  (class class class)co 7431  Topctop 22899  clsccl 23026  Filcfil 23853   fClus cfcls 23944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-fil 23854  df-fcls 23949
This theorem is referenced by:  fclstopon  24020  fclsneii  24025  fclsfnflim  24035  flimfnfcls  24036
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