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

Theorem fclsnei 23844
Description: Cluster points in terms of neighborhoods. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclsnei ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅)))
Distinct variable groups:   𝑛,𝑠,𝐴   𝑛,𝐹,𝑠   𝑛,𝐽,𝑠   𝑋,𝑠
Allowed substitution hint:   𝑋(𝑛)

Proof of Theorem fclsnei
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 eqid 2724 . . . . 5 𝐽 = 𝐽
21fclselbas 23841 . . . 4 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 𝐽)
3 toponuni 22737 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
43adantr 480 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝑋 = 𝐽)
54eleq2d 2811 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴𝑋𝐴 𝐽))
62, 5imbitrrid 245 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴𝑋))
7 fclsneii 23842 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑠𝐹) → (𝑛𝑠) ≠ ∅)
873expb 1117 . . . 4 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑠𝐹)) → (𝑛𝑠) ≠ ∅)
98ralrimivva 3192 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅)
106, 9jca2 513 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅)))
11 topontop 22736 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1211ad3antrrr 727 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → 𝐽 ∈ Top)
13 simprl 768 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → 𝑜𝐽)
14 simprr 770 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → 𝐴𝑜)
15 opnneip 22944 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝐽𝐴𝑜) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
1612, 13, 14, 15syl3anc 1368 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
17 ineq1 4197 . . . . . . . . . . 11 (𝑛 = 𝑜 → (𝑛𝑠) = (𝑜𝑠))
1817neeq1d 2992 . . . . . . . . . 10 (𝑛 = 𝑜 → ((𝑛𝑠) ≠ ∅ ↔ (𝑜𝑠) ≠ ∅))
1918ralbidv 3169 . . . . . . . . 9 (𝑛 = 𝑜 → (∀𝑠𝐹 (𝑛𝑠) ≠ ∅ ↔ ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
2019rspcv 3600 . . . . . . . 8 (𝑜 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅ → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
2116, 20syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅ → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
2221expr 456 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑜𝐽) → (𝐴𝑜 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅ → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
2322com23 86 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑜𝐽) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅ → (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
2423ralrimdva 3146 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅ → ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
2524imdistanda 571 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅) → (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
26 fclsopn 23839 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
2725, 26sylibrd 259 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅) → 𝐴 ∈ (𝐽 fClus 𝐹)))
2810, 27impbid 211 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wne 2932  wral 3053  cin 3939  c0 4314  {csn 4620   cuni 4899  cfv 6533  (class class class)co 7401  Topctop 22716  TopOnctopon 22733  neicnei 22922  Filcfil 23670   fClus cfcls 23761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-fbas 21224  df-top 22717  df-topon 22734  df-cld 22844  df-ntr 22845  df-cls 22846  df-nei 22923  df-fil 23671  df-fcls 23766
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator