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Theorem fclsnei 21733
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 2621 . . . . 5 𝐽 = 𝐽
21fclselbas 21730 . . . 4 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 𝐽)
3 toponuni 20642 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
43adantr 481 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝑋 = 𝐽)
54eleq2d 2684 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴𝑋𝐴 𝐽))
62, 5syl5ibr 236 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴𝑋))
7 fclsneii 21731 . . . . . 6 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑠𝐹) → (𝑛𝑠) ≠ ∅)
873expb 1263 . . . . 5 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑠𝐹)) → (𝑛𝑠) ≠ ∅)
98ralrimivva 2965 . . . 4 (𝐴 ∈ (𝐽 fClus 𝐹) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅)
109a1i 11 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅))
116, 10jcad 555 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅)))
12 topontop 20641 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1312ad3antrrr 765 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → 𝐽 ∈ Top)
14 simprl 793 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → 𝑜𝐽)
15 simprr 795 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → 𝐴𝑜)
16 opnneip 20833 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝐽𝐴𝑜) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
1713, 14, 15, 16syl3anc 1323 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
18 ineq1 3785 . . . . . . . . . . 11 (𝑛 = 𝑜 → (𝑛𝑠) = (𝑜𝑠))
1918neeq1d 2849 . . . . . . . . . 10 (𝑛 = 𝑜 → ((𝑛𝑠) ≠ ∅ ↔ (𝑜𝑠) ≠ ∅))
2019ralbidv 2980 . . . . . . . . 9 (𝑛 = 𝑜 → (∀𝑠𝐹 (𝑛𝑠) ≠ ∅ ↔ ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
2120rspcv 3291 . . . . . . . 8 (𝑜 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅ → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
2217, 21syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅ → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
2322expr 642 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑜𝐽) → (𝐴𝑜 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅ → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
2423com23 86 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑜𝐽) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅ → (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
2524ralrimdva 2963 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅ → ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
2625imdistanda 728 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅) → (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
27 fclsopn 21728 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
2826, 27sylibrd 249 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅) → 𝐴 ∈ (𝐽 fClus 𝐹)))
2911, 28impbid 202 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 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  cin 3554  c0 3891  {csn 4148   cuni 4402  cfv 5847  (class class class)co 6604  Topctop 20617  TopOnctopon 20618  neicnei 20811  Filcfil 21559   fClus cfcls 21650
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-fbas 19662  df-top 20621  df-topon 20623  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-fil 21560  df-fcls 21655
This theorem is referenced by: (None)
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