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Theorem fclsopni 22053
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 2817 . . . . . . . . 9 𝐽 = 𝐽
21fclsfil 22048 . . . . . . . 8 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
3 fclstopon 22050 . . . . . . . 8 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘ 𝐽) ↔ 𝐹 ∈ (Fil‘ 𝐽)))
42, 3mpbird 248 . . . . . . 7 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ (TopOn‘ 𝐽))
5 fclsopn 22052 . . . . . . 7 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹 ∈ (Fil‘ 𝐽)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 𝐽 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
64, 2, 5syl2anc 575 . . . . . 6 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 𝐽 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))
76ibi 258 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 𝐽 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅)))
87simprd 485 . . . 4 (𝐴 ∈ (𝐽 fClus 𝐹) → ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))
9 eleq2 2885 . . . . . 6 (𝑜 = 𝑈 → (𝐴𝑜𝐴𝑈))
10 ineq1 4017 . . . . . . . 8 (𝑜 = 𝑈 → (𝑜𝑠) = (𝑈𝑠))
1110neeq1d 3048 . . . . . . 7 (𝑜 = 𝑈 → ((𝑜𝑠) ≠ ∅ ↔ (𝑈𝑠) ≠ ∅))
1211ralbidv 3185 . . . . . 6 (𝑜 = 𝑈 → (∀𝑠𝐹 (𝑜𝑠) ≠ ∅ ↔ ∀𝑠𝐹 (𝑈𝑠) ≠ ∅))
139, 12imbi12d 335 . . . . 5 (𝑜 = 𝑈 → ((𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅) ↔ (𝐴𝑈 → ∀𝑠𝐹 (𝑈𝑠) ≠ ∅)))
1413rspccv 3510 . . . 4 (∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅) → (𝑈𝐽 → (𝐴𝑈 → ∀𝑠𝐹 (𝑈𝑠) ≠ ∅)))
158, 14syl 17 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑈𝐽 → (𝐴𝑈 → ∀𝑠𝐹 (𝑈𝑠) ≠ ∅)))
16 ineq2 4018 . . . . 5 (𝑠 = 𝑆 → (𝑈𝑠) = (𝑈𝑆))
1716neeq1d 3048 . . . 4 (𝑠 = 𝑆 → ((𝑈𝑠) ≠ ∅ ↔ (𝑈𝑆) ≠ ∅))
1817rspccv 3510 . . 3 (∀𝑠𝐹 (𝑈𝑠) ≠ ∅ → (𝑆𝐹 → (𝑈𝑆) ≠ ∅))
1915, 18syl8 76 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑈𝐽 → (𝐴𝑈 → (𝑆𝐹 → (𝑈𝑆) ≠ ∅))))
20193imp2 1451 1 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈𝐽𝐴𝑈𝑆𝐹)) → (𝑈𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  wne 2989  wral 3107  cin 3779  c0 4127   cuni 4641  cfv 6111  (class class class)co 6884  TopOnctopon 20949  Filcfil 21883   fClus cfcls 21974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-iin 4726  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-fbas 19971  df-top 20933  df-topon 20950  df-cld 21058  df-ntr 21059  df-cls 21060  df-fil 21884  df-fcls 21979
This theorem is referenced by:  fclsneii  22055  supnfcls  22058  flimfnfcls  22066  cfilfcls  23306
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