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Theorem fclstop 22725
 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 2758 . . 3 𝐽 = 𝐽
21isfcls 22723 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
32simp1bi 1142 1 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  ∀wral 3070  ∪ cuni 4801  ‘cfv 6340  (class class class)co 7156  Topctop 21607  clsccl 21732  Filcfil 22559   fClus cfcls 22650 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 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-iin 4889  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-fbas 20177  df-fil 22560  df-fcls 22655 This theorem is referenced by:  fclstopon  22726  fclsneii  22731  fclsfnflim  22741  flimfnfcls  22742
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