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Theorem fclsss1 22245
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 1203 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐽𝐾)
2 ssralv 3885 . . . . . . 7 (𝐽𝐾 → (∀𝑜𝐾 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅) → ∀𝑜𝐽 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
32anim2d 605 . . . . . 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 1201 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
6 fclstopon 22235 . . . . . . . 8 (𝑥 ∈ (𝐾 fClus 𝐹) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
76adantl 475 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
85, 7mpbird 249 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐾 ∈ (TopOn‘𝑋))
9 fclsopn 22237 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐾 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑜𝐾 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
108, 5, 9syl2anc 579 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝑥 ∈ (𝐾 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑜𝐾 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
11 simpl1 1199 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐽 ∈ (TopOn‘𝑋))
12 fclsopn 22237 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑜𝐽 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
1311, 5, 12syl2anc 579 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑜𝐽 (𝑥𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
144, 10, 133imtr4d 286 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝑥 ∈ (𝐾 fClus 𝐹) → 𝑥 ∈ (𝐽 fClus 𝐹)))
1514ex 403 . . 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 3827 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 198  wa 386  w3a 1071  wcel 2107  wne 2969  wral 3090  cin 3791  wss 3792  c0 4141  cfv 6137  (class class class)co 6924  TopOnctopon 21133  Filcfil 22068   fClus cfcls 22159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-fbas 20150  df-top 21117  df-topon 21134  df-cld 21242  df-ntr 21243  df-cls 21244  df-fil 22069  df-fcls 22164
This theorem is referenced by:  fclscf  22248
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