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Theorem fclsopn 21758
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 21757 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
2 filn0 21606 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
32adantl 482 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ≠ ∅)
4 r19.2z 4038 . . . . . 6 ((𝐹 ≠ ∅ ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
54ex 450 . . . . 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 20658 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
87ad2antrr 761 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝐽 ∈ Top)
9 filelss 21596 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → 𝑠𝑋)
109adantll 749 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝑠𝑋)
11 toponuni 20659 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1211ad2antrr 761 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝑋 = 𝐽)
1310, 12sseqtrd 3626 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝑠 𝐽)
14 eqid 2621 . . . . . . . . 9 𝐽 = 𝐽
1514clsss3 20803 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑠 𝐽) → ((cls‘𝐽)‘𝑠) ⊆ 𝐽)
168, 13, 15syl2anc 692 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → ((cls‘𝐽)‘𝑠) ⊆ 𝐽)
1716, 12sseqtr4d 3627 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → ((cls‘𝐽)‘𝑠) ⊆ 𝑋)
1817sseld 3587 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴𝑋))
1918rexlimdva 3026 . . . 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 666 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ (𝐴𝑋 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
227ad3antrrr 765 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝐽 ∈ Top)
2313adantlr 750 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝑠 𝐽)
24 simplr 791 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝐴𝑋)
2511ad3antrrr 765 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝑋 = 𝐽)
2624, 25eleqtrd 2700 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝐴 𝐽)
2714elcls 20817 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑠 𝐽𝐴 𝐽) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅)))
2822, 23, 26, 27syl3anc 1323 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅)))
2928ralbidva 2981 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑠𝐹𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅)))
30 ralcom 3092 . . . . 5 (∀𝑠𝐹𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ ∀𝑜𝐽𝑠𝐹 (𝐴𝑜 → (𝑜𝑠) ≠ ∅))
31 r19.21v 2956 . . . . . 6 (∀𝑠𝐹 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
3231ralbii 2976 . . . . 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 672 . 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 384   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  cin 3559  wss 3560  c0 3897   cuni 4409  cfv 5857  (class class class)co 6615  Topctop 20638  TopOnctopon 20655  clsccl 20762  Filcfil 21589   fClus cfcls 21680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-fbas 19683  df-top 20639  df-topon 20656  df-cld 20763  df-ntr 20764  df-cls 20765  df-fil 21590  df-fcls 21685
This theorem is referenced by:  fclsopni  21759  fclselbas  21760  fclsnei  21763  fclsbas  21765  fclsss1  21766  fclsrest  21768  fclscf  21769  isfcf  21778
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