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

Theorem fclsopn 22538
Description: Write the cluster point condition in terms of open sets. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
fclsopn ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
Distinct variable groups:   𝑜,𝑠,𝐴   𝑜,𝐹,𝑠   𝑜,𝐽,𝑠   𝑜,𝑋,𝑠

Proof of Theorem fclsopn
StepHypRef Expression
1 isfcls2 22537 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
2 filn0 22386 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
32adantl 482 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ≠ ∅)
4 r19.2z 4442 . . . . . 6 ((𝐹 ≠ ∅ ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
54ex 413 . . . . 5 (𝐹 ≠ ∅ → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
63, 5syl 17 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
7 topontop 21437 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
87ad2antrr 722 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝐽 ∈ Top)
9 filelss 22376 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → 𝑠𝑋)
109adantll 710 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝑠𝑋)
11 toponuni 21438 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1211ad2antrr 722 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝑋 = 𝐽)
1310, 12sseqtrd 4010 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝑠 𝐽)
14 eqid 2825 . . . . . . . . 9 𝐽 = 𝐽
1514clsss3 21583 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑠 𝐽) → ((cls‘𝐽)‘𝑠) ⊆ 𝐽)
168, 13, 15syl2anc 584 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → ((cls‘𝐽)‘𝑠) ⊆ 𝐽)
1716, 12sseqtrrd 4011 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → ((cls‘𝐽)‘𝑠) ⊆ 𝑋)
1817sseld 3969 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴𝑋))
1918rexlimdva 3288 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴𝑋))
206, 19syld 47 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴𝑋))
2120pm4.71rd 563 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ (𝐴𝑋 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
227ad3antrrr 726 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝐽 ∈ Top)
2313adantlr 711 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝑠 𝐽)
24 simplr 765 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝐴𝑋)
2511ad3antrrr 726 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝑋 = 𝐽)
2624, 25eleqtrd 2919 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝐴 𝐽)
2714elcls 21597 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑠 𝐽𝐴 𝐽) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅)))
2822, 23, 26, 27syl3anc 1365 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅)))
2928ralbidva 3200 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑠𝐹𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅)))
30 ralcom 3358 . . . . 5 (∀𝑠𝐹𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ ∀𝑜𝐽𝑠𝐹 (𝐴𝑜 → (𝑜𝑠) ≠ ∅))
31 r19.21v 3179 . . . . . 6 (∀𝑠𝐹 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
3231ralbii 3169 . . . . 5 (∀𝑜𝐽𝑠𝐹 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
3330, 32bitri 276 . . . 4 (∀𝑠𝐹𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
3429, 33syl6bb 288 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
3534pm5.32da 579 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴𝑋 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
361, 21, 353bitrd 306 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3020  wral 3142  wrex 3143  cin 3938  wss 3939  c0 4294   cuni 4836  cfv 6351  (class class class)co 7151  Topctop 21417  TopOnctopon 21434  clsccl 21542  Filcfil 22369   fClus cfcls 22460
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-fbas 20458  df-top 21418  df-topon 21435  df-cld 21543  df-ntr 21544  df-cls 21545  df-fil 22370  df-fcls 22465
This theorem is referenced by:  fclsopni  22539  fclselbas  22540  fclsnei  22543  fclsbas  22545  fclsss1  22546  fclsrest  22548  fclscf  22549  isfcf  22558
  Copyright terms: Public domain W3C validator