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Theorem fclssscls 21632
 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 2610 . . . . 5 𝐽 = 𝐽
21isfcls 21623 . . . 4 (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽) ∧ ∀𝑠𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
32simp3bi 1071 . . 3 (𝑥 ∈ (𝐽 fClus 𝐹) → ∀𝑠𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠))
4 fveq2 6103 . . . . 5 (𝑠 = 𝑆 → ((cls‘𝐽)‘𝑠) = ((cls‘𝐽)‘𝑆))
54eleq2d 2673 . . . 4 (𝑠 = 𝑆 → (𝑥 ∈ ((cls‘𝐽)‘𝑠) ↔ 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
65rspcv 3278 . . 3 (𝑆𝐹 → (∀𝑠𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
73, 6syl5 33 . 2 (𝑆𝐹 → (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
87ssrdv 3574 1 (𝑆𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549  Topctop 20517  clsccl 20632  Filcfil 21459   fClus cfcls 21550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-fbas 19564  df-fil 21460  df-fcls 21555 This theorem is referenced by:  fclscmp  21644
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