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Theorem fclsss2 21637
 Description: A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
fclsss2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝐽 fClus 𝐺) ⊆ (𝐽 fClus 𝐹))

Proof of Theorem fclsss2
Dummy variables 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 1059 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐹𝐺)
2 ssralv 3629 . . . . . 6 (𝐹𝐺 → (∀𝑠𝐺 𝑥 ∈ ((cls‘𝐽)‘𝑠) → ∀𝑠𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
31, 2syl 17 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → (∀𝑠𝐺 𝑥 ∈ ((cls‘𝐽)‘𝑠) → ∀𝑠𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
4 simpl1 1057 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐽 ∈ (TopOn‘𝑋))
5 fclstopon 21626 . . . . . . . 8 (𝑥 ∈ (𝐽 fClus 𝐺) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐺 ∈ (Fil‘𝑋)))
65adantl 481 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐺 ∈ (Fil‘𝑋)))
74, 6mpbid 221 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐺 ∈ (Fil‘𝑋))
8 isfcls2 21627 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐺) ↔ ∀𝑠𝐺 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
94, 7, 8syl2anc 691 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → (𝑥 ∈ (𝐽 fClus 𝐺) ↔ ∀𝑠𝐺 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
10 simpl2 1058 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐹 ∈ (Fil‘𝑋))
11 isfcls2 21627 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
124, 10, 11syl2anc 691 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
133, 9, 123imtr4d 282 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → (𝑥 ∈ (𝐽 fClus 𝐺) → 𝑥 ∈ (𝐽 fClus 𝐹)))
1413ex 449 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥 ∈ (𝐽 fClus 𝐺) → (𝑥 ∈ (𝐽 fClus 𝐺) → 𝑥 ∈ (𝐽 fClus 𝐹))))
1514pm2.43d 51 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝑥 ∈ (𝐽 fClus 𝐺) → 𝑥 ∈ (𝐽 fClus 𝐹)))
1615ssrdv 3574 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝐽 fClus 𝐺) ⊆ (𝐽 fClus 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  TopOnctopon 20518  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  ax-un 6847 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-topon 20523  df-fil 21460  df-fcls 21555 This theorem is referenced by:  fclsfnflim  21641  ufilcmp  21646  cnpfcfi  21654
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