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

Theorem fclsopn 22037
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 22036 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
2 filn0 21885 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
32adantl 467 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ≠ ∅)
4 r19.2z 4199 . . . . . 6 ((𝐹 ≠ ∅ ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
54ex 397 . . . . 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 20937 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
87ad2antrr 697 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝐽 ∈ Top)
9 filelss 21875 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → 𝑠𝑋)
109adantll 685 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝑠𝑋)
11 toponuni 20938 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1211ad2antrr 697 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝑋 = 𝐽)
1310, 12sseqtrd 3788 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝑠 𝐽)
14 eqid 2770 . . . . . . . . 9 𝐽 = 𝐽
1514clsss3 21083 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑠 𝐽) → ((cls‘𝐽)‘𝑠) ⊆ 𝐽)
168, 13, 15syl2anc 565 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → ((cls‘𝐽)‘𝑠) ⊆ 𝐽)
1716, 12sseqtr4d 3789 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → ((cls‘𝐽)‘𝑠) ⊆ 𝑋)
1817sseld 3749 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴𝑋))
1918rexlimdva 3178 . . . 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 544 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ (𝐴𝑋 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
227ad3antrrr 701 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝐽 ∈ Top)
2313adantlr 686 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝑠 𝐽)
24 simplr 744 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝐴𝑋)
2511ad3antrrr 701 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝑋 = 𝐽)
2624, 25eleqtrd 2851 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝐴 𝐽)
2714elcls 21097 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑠 𝐽𝐴 𝐽) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅)))
2822, 23, 26, 27syl3anc 1475 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅)))
2928ralbidva 3133 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑠𝐹𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅)))
30 ralcom 3245 . . . . 5 (∀𝑠𝐹𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ ∀𝑜𝐽𝑠𝐹 (𝐴𝑜 → (𝑜𝑠) ≠ ∅))
31 r19.21v 3108 . . . . . 6 (∀𝑠𝐹 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
3231ralbii 3128 . . . . 5 (∀𝑜𝐽𝑠𝐹 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
3330, 32bitri 264 . . . 4 (∀𝑠𝐹𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
3429, 33syl6bb 276 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
3534pm5.32da 560 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴𝑋 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
361, 21, 353bitrd 294 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 196  wa 382   = wceq 1630  wcel 2144  wne 2942  wral 3060  wrex 3061  cin 3720  wss 3721  c0 4061   cuni 4572  cfv 6031  (class class class)co 6792  Topctop 20917  TopOnctopon 20934  clsccl 21042  Filcfil 21868   fClus cfcls 21959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-fbas 19957  df-top 20918  df-topon 20935  df-cld 21043  df-ntr 21044  df-cls 21045  df-fil 21869  df-fcls 21964
This theorem is referenced by:  fclsopni  22038  fclselbas  22039  fclsnei  22042  fclsbas  22044  fclsss1  22045  fclsrest  22047  fclscf  22048  isfcf  22057
  Copyright terms: Public domain W3C validator