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Theorem fclsss1 22630
Description: A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclsss1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝐾 fClus 𝐹) ⊆ (𝐽 fClus 𝐹))

Proof of Theorem fclsss1
Dummy variables 𝑜 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 1190 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐽𝐾)
2 ssralv 4019 . . . . . . 7 (𝐽𝐾 → (∀𝑜𝐾 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅) → ∀𝑜𝐽 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
32anim2d 614 . . . . . 6 (𝐽𝐾 → ((𝑥𝑋 ∧ ∀𝑜𝐾 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)) → (𝑥𝑋 ∧ ∀𝑜𝐽 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
41, 3syl 17 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → ((𝑥𝑋 ∧ ∀𝑜𝐾 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)) → (𝑥𝑋 ∧ ∀𝑜𝐽 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
5 simpl2 1189 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
6 fclstopon 22620 . . . . . . . 8 (𝑥 ∈ (𝐾 fClus 𝐹) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
76adantl 485 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
85, 7mpbird 260 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐾 ∈ (TopOn‘𝑋))
9 fclsopn 22622 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐾 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑜𝐾 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
108, 5, 9syl2anc 587 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝑥 ∈ (𝐾 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑜𝐾 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
11 simpl1 1188 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐽 ∈ (TopOn‘𝑋))
12 fclsopn 22622 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑜𝐽 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
1311, 5, 12syl2anc 587 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑜𝐽 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
144, 10, 133imtr4d 297 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝑥 ∈ (𝐾 fClus 𝐹) → 𝑥 ∈ (𝐽 fClus 𝐹)))
1514ex 416 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝐾 fClus 𝐹) → (𝑥 ∈ (𝐾 fClus 𝐹) → 𝑥 ∈ (𝐽 fClus 𝐹))))
1615pm2.43d 53 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝐾 fClus 𝐹) → 𝑥 ∈ (𝐽 fClus 𝐹)))
1716ssrdv 3959 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝐾 fClus 𝐹) ⊆ (𝐽 fClus 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wcel 2115  wne 3014  wral 3133  cin 3918  wss 3919  c0 4276  cfv 6343  (class class class)co 7149  TopOnctopon 21518  Filcfil 22453   fClus cfcls 22544
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-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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-reu 3140  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 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  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-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-fbas 20542  df-top 21502  df-topon 21519  df-cld 21627  df-ntr 21628  df-cls 21629  df-fil 22454  df-fcls 22549
This theorem is referenced by:  fclscf  22633
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