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Theorem fcfneii 22621
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 22619 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅)))
2 ineq1 4156 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 ∩ (𝐹𝑠)) = (𝑁 ∩ (𝐹𝑠)))
32neeq1d 3066 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹𝑠)) ≠ ∅))
4 imaeq2 5898 . . . . . . . . 9 (𝑠 = 𝑆 → (𝐹𝑠) = (𝐹𝑆))
54ineq2d 4164 . . . . . . . 8 (𝑠 = 𝑆 → (𝑁 ∩ (𝐹𝑠)) = (𝑁 ∩ (𝐹𝑆)))
65neeq1d 3066 . . . . . . 7 (𝑠 = 𝑆 → ((𝑁 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹𝑆)) ≠ ∅))
73, 6rspc2v 3610 . . . . . 6 ((𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐿) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹𝑆)) ≠ ∅))
87ex 416 . . . . 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 485 . . 3 ((𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)))
111, 10syl6bi 256 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (𝑁 ∩ (𝐹𝑆)) ≠ ∅))))
12113imp2 1346 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 399  w3a 1084   = wceq 1538  wcel 2115  wne 3007  wral 3126  cin 3909  c0 4266  {csn 4540  cima 5531  wf 6324  cfv 6328  (class class class)co 7130  TopOnctopon 21494  neicnei 21681  Filcfil 22429   fClusf cfcf 22521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-fbas 20518  df-fg 20519  df-top 21478  df-topon 21495  df-cld 21603  df-ntr 21604  df-cls 21605  df-nei 21682  df-fil 22430  df-fm 22522  df-fcls 22525  df-fcf 22526
This theorem is referenced by: (None)
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