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Theorem fcfneii 21751
Description: A neighborhood of a cluster point of a function contains a function value from every tail. (Contributed by Jeff Hankins, 27-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
fcfneii (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐿)) → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)

Proof of Theorem fcfneii
Dummy variables 𝑛 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fcfnei 21749 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅)))
2 ineq1 3785 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 ∩ (𝐹𝑠)) = (𝑁 ∩ (𝐹𝑠)))
32neeq1d 2849 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹𝑠)) ≠ ∅))
4 imaeq2 5421 . . . . . . . . 9 (𝑠 = 𝑆 → (𝐹𝑠) = (𝐹𝑆))
54ineq2d 3792 . . . . . . . 8 (𝑠 = 𝑆 → (𝑁 ∩ (𝐹𝑠)) = (𝑁 ∩ (𝐹𝑆)))
65neeq1d 2849 . . . . . . 7 (𝑠 = 𝑆 → ((𝑁 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹𝑆)) ≠ ∅))
73, 6rspc2v 3306 . . . . . 6 ((𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐿) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹𝑆)) ≠ ∅))
87ex 450 . . . . 5 (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)))
98com3r 87 . . . 4 (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)))
109adantl 482 . . 3 ((𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)))
111, 10syl6bi 243 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (𝑁 ∩ (𝐹𝑆)) ≠ ∅))))
12113imp2 1279 1 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐿)) → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  cin 3554  c0 3891  {csn 4148  cima 5077  wf 5843  cfv 5847  (class class class)co 6604  TopOnctopon 20618  neicnei 20811  Filcfil 21559   fClusf cfcf 21651
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-map 7804  df-fbas 19662  df-fg 19663  df-top 20621  df-topon 20623  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-fil 21560  df-fm 21652  df-fcls 21655  df-fcf 21656
This theorem is referenced by: (None)
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