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Theorem fclsopn 22143
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 22142 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
2 filn0 21991 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
32adantl 474 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ≠ ∅)
4 r19.2z 4251 . . . . . 6 ((𝐹 ≠ ∅ ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
54ex 402 . . . . 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 21043 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
87ad2antrr 718 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝐽 ∈ Top)
9 filelss 21981 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → 𝑠𝑋)
109adantll 706 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝑠𝑋)
11 toponuni 21044 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1211ad2antrr 718 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝑋 = 𝐽)
1310, 12sseqtrd 3835 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → 𝑠 𝐽)
14 eqid 2797 . . . . . . . . 9 𝐽 = 𝐽
1514clsss3 21189 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑠 𝐽) → ((cls‘𝐽)‘𝑠) ⊆ 𝐽)
168, 13, 15syl2anc 580 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → ((cls‘𝐽)‘𝑠) ⊆ 𝐽)
1716, 12sseqtr4d 3836 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → ((cls‘𝐽)‘𝑠) ⊆ 𝑋)
1817sseld 3795 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴𝑋))
1918rexlimdva 3210 . . . 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 559 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ (𝐴𝑋 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
227ad3antrrr 722 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝐽 ∈ Top)
2313adantlr 707 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝑠 𝐽)
24 simplr 786 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝐴𝑋)
2511ad3antrrr 722 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝑋 = 𝐽)
2624, 25eleqtrd 2878 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → 𝐴 𝐽)
2714elcls 21203 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑠 𝐽𝐴 𝐽) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅)))
2822, 23, 26, 27syl3anc 1491 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑠𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅)))
2928ralbidva 3164 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑠𝐹𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅)))
30 ralcom 3277 . . . . 5 (∀𝑠𝐹𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ ∀𝑜𝐽𝑠𝐹 (𝐴𝑜 → (𝑜𝑠) ≠ ∅))
31 r19.21v 3139 . . . . . 6 (∀𝑠𝐹 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
3231ralbii 3159 . . . . 5 (∀𝑜𝐽𝑠𝐹 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
3330, 32bitri 267 . . . 4 (∀𝑠𝐹𝑜𝐽 (𝐴𝑜 → (𝑜𝑠) ≠ ∅) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
3429, 33syl6bb 279 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
3534pm5.32da 575 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴𝑋 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
361, 21, 353bitrd 297 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 198  wa 385   = wceq 1653  wcel 2157  wne 2969  wral 3087  wrex 3088  cin 3766  wss 3767  c0 4113   cuni 4626  cfv 6099  (class class class)co 6876  Topctop 21023  TopOnctopon 21040  clsccl 21148  Filcfil 21974   fClus cfcls 22065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-iin 4711  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-fbas 20062  df-top 21024  df-topon 21041  df-cld 21149  df-ntr 21150  df-cls 21151  df-fil 21975  df-fcls 22070
This theorem is referenced by:  fclsopni  22144  fclselbas  22145  fclsnei  22148  fclsbas  22150  fclsss1  22151  fclsrest  22153  fclscf  22154  isfcf  22163
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