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Theorem fclssscls 22619
Description: The set of cluster points is a subset of the closure of any filter element. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclssscls (𝑆𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem fclssscls
Dummy variables 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . . . 5 𝐽 = 𝐽
21isfcls 22610 . . . 4 (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽) ∧ ∀𝑠𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
32simp3bi 1144 . . 3 (𝑥 ∈ (𝐽 fClus 𝐹) → ∀𝑠𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠))
4 fveq2 6658 . . . . 5 (𝑠 = 𝑆 → ((cls‘𝐽)‘𝑠) = ((cls‘𝐽)‘𝑆))
54eleq2d 2901 . . . 4 (𝑠 = 𝑆 → (𝑥 ∈ ((cls‘𝐽)‘𝑠) ↔ 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
65rspcv 3604 . . 3 (𝑆𝐹 → (∀𝑠𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
73, 6syl5 34 . 2 (𝑆𝐹 → (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
87ssrdv 3958 1 (𝑆𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wral 3133  wss 3919   cuni 4824  cfv 6343  (class class class)co 7145  Topctop 21494  clsccl 21619  Filcfil 22446   fClus cfcls 22537
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-int 4863  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-fbas 20535  df-fil 22447  df-fcls 22542
This theorem is referenced by:  fclscmp  22631
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