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Theorem fclsopni 22020
Description: An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclsopni ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈𝐽𝐴𝑈𝑆𝐹)) → (𝑈𝑆) ≠ ∅)

Proof of Theorem fclsopni
Dummy variables 𝑜 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2760 . . . . . . . . 9 𝐽 = 𝐽
21fclsfil 22015 . . . . . . . 8 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
3 fclstopon 22017 . . . . . . . 8 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘ 𝐽) ↔ 𝐹 ∈ (Fil‘ 𝐽)))
42, 3mpbird 247 . . . . . . 7 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ (TopOn‘ 𝐽))
5 fclsopn 22019 . . . . . . 7 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹 ∈ (Fil‘ 𝐽)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 𝐽 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
64, 2, 5syl2anc 696 . . . . . 6 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 𝐽 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
76ibi 256 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 𝐽 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
87simprd 482 . . . 4 (𝐴 ∈ (𝐽 fClus 𝐹) → ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
9 eleq2 2828 . . . . . 6 (𝑜 = 𝑈 → (𝐴𝑜𝐴𝑈))
10 ineq1 3950 . . . . . . . 8 (𝑜 = 𝑈 → (𝑜𝑠) = (𝑈𝑠))
1110neeq1d 2991 . . . . . . 7 (𝑜 = 𝑈 → ((𝑜𝑠) ≠ ∅ ↔ (𝑈𝑠) ≠ ∅))
1211ralbidv 3124 . . . . . 6 (𝑜 = 𝑈 → (∀𝑠𝐹 (𝑜𝑠) ≠ ∅ ↔ ∀𝑠𝐹 (𝑈𝑠) ≠ ∅))
139, 12imbi12d 333 . . . . 5 (𝑜 = 𝑈 → ((𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅) ↔ (𝐴𝑈 → ∀𝑠𝐹 (𝑈𝑠) ≠ ∅)))
1413rspccv 3446 . . . 4 (∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅) → (𝑈𝐽 → (𝐴𝑈 → ∀𝑠𝐹 (𝑈𝑠) ≠ ∅)))
158, 14syl 17 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑈𝐽 → (𝐴𝑈 → ∀𝑠𝐹 (𝑈𝑠) ≠ ∅)))
16 ineq2 3951 . . . . 5 (𝑠 = 𝑆 → (𝑈𝑠) = (𝑈𝑆))
1716neeq1d 2991 . . . 4 (𝑠 = 𝑆 → ((𝑈𝑠) ≠ ∅ ↔ (𝑈𝑆) ≠ ∅))
1817rspccv 3446 . . 3 (∀𝑠𝐹 (𝑈𝑠) ≠ ∅ → (𝑆𝐹 → (𝑈𝑆) ≠ ∅))
1915, 18syl8 76 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑈𝐽 → (𝐴𝑈 → (𝑆𝐹 → (𝑈𝑆) ≠ ∅))))
20193imp2 1443 1 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈𝐽𝐴𝑈𝑆𝐹)) → (𝑈𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  cin 3714  c0 4058   cuni 4588  cfv 6049  (class class class)co 6813  TopOnctopon 20917  Filcfil 21850   fClus cfcls 21941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-fbas 19945  df-top 20901  df-topon 20918  df-cld 21025  df-ntr 21026  df-cls 21027  df-fil 21851  df-fcls 21946
This theorem is referenced by:  fclsneii  22022  supnfcls  22025  flimfnfcls  22033  cfilfcls  23272
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