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Theorem fclssscls 24042
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 2735 . . . . 5 𝐽 = 𝐽
21isfcls 24033 . . . 4 (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽) ∧ ∀𝑠𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
32simp3bi 1146 . . 3 (𝑥 ∈ (𝐽 fClus 𝐹) → ∀𝑠𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠))
4 fveq2 6907 . . . . 5 (𝑠 = 𝑆 → ((cls‘𝐽)‘𝑠) = ((cls‘𝐽)‘𝑆))
54eleq2d 2825 . . . 4 (𝑠 = 𝑆 → (𝑥 ∈ ((cls‘𝐽)‘𝑠) ↔ 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
65rspcv 3618 . . 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 4001 1 (𝑆𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  wss 3963   cuni 4912  cfv 6563  (class class class)co 7431  Topctop 22915  clsccl 23042  Filcfil 23869   fClus cfcls 23960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-fil 23870  df-fcls 23965
This theorem is referenced by:  fclscmp  24054
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