Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fclsss1 Structured version   Visualization version   GIF version

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
 Copyright terms: Public domain W3C validator