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Theorem fcfneii 21651
 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 21649 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅)))
2 ineq1 3769 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 ∩ (𝐹𝑠)) = (𝑁 ∩ (𝐹𝑠)))
32neeq1d 2841 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹𝑠)) ≠ ∅))
4 imaeq2 5381 . . . . . . . . 9 (𝑠 = 𝑆 → (𝐹𝑠) = (𝐹𝑆))
54ineq2d 3776 . . . . . . . 8 (𝑠 = 𝑆 → (𝑁 ∩ (𝐹𝑠)) = (𝑁 ∩ (𝐹𝑆)))
65neeq1d 2841 . . . . . . 7 (𝑠 = 𝑆 → ((𝑁 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹𝑆)) ≠ ∅))
73, 6rspc2v 3293 . . . . . 6 ((𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐿) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹𝑆)) ≠ ∅))
87ex 449 . . . . 5 (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)))
98com3r 85 . . . 4 (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)))
109adantl 481 . . 3 ((𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)))
111, 10syl6bi 242 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (𝑁 ∩ (𝐹𝑆)) ≠ ∅))))
12113imp2 1274 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ∩ cin 3539  ∅c0 3874  {csn 4125   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  TopOnctopon 20518  neicnei 20711  Filcfil 21459   fClusf cfcf 21551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-fbas 19564  df-fg 19565  df-top 20521  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-fil 21460  df-fm 21552  df-fcls 21555  df-fcf 21556 This theorem is referenced by: (None)
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