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Theorem fcfneii 22249
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 22247 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅)))
2 ineq1 4030 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 ∩ (𝐹𝑠)) = (𝑁 ∩ (𝐹𝑠)))
32neeq1d 3028 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹𝑠)) ≠ ∅))
4 imaeq2 5716 . . . . . . . . 9 (𝑠 = 𝑆 → (𝐹𝑠) = (𝐹𝑆))
54ineq2d 4037 . . . . . . . 8 (𝑠 = 𝑆 → (𝑁 ∩ (𝐹𝑠)) = (𝑁 ∩ (𝐹𝑆)))
65neeq1d 3028 . . . . . . 7 (𝑠 = 𝑆 → ((𝑁 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹𝑆)) ≠ ∅))
73, 6rspc2v 3524 . . . . . 6 ((𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐿) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹𝑆)) ≠ ∅))
87ex 403 . . . . 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 475 . . 3 ((𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)))
111, 10syl6bi 245 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (𝑁 ∩ (𝐹𝑆)) ≠ ∅))))
12113imp2 1411 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 386  w3a 1071   = wceq 1601  wcel 2107  wne 2969  wral 3090  cin 3791  c0 4141  {csn 4398  cima 5358  wf 6131  cfv 6135  (class class class)co 6922  TopOnctopon 21122  neicnei 21309  Filcfil 22057   fClusf cfcf 22149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-fbas 20139  df-fg 20140  df-top 21106  df-topon 21123  df-cld 21231  df-ntr 21232  df-cls 21233  df-nei 21310  df-fil 22058  df-fm 22150  df-fcls 22153  df-fcf 22154
This theorem is referenced by: (None)
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