MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fcfneii Structured version   Visualization version   GIF version

Theorem fcfneii 24061
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 24059 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅)))
2 ineq1 4221 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 ∩ (𝐹𝑠)) = (𝑁 ∩ (𝐹𝑠)))
32neeq1d 2998 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹𝑠)) ≠ ∅))
4 imaeq2 6076 . . . . . . . . 9 (𝑠 = 𝑆 → (𝐹𝑠) = (𝐹𝑆))
54ineq2d 4228 . . . . . . . 8 (𝑠 = 𝑆 → (𝑁 ∩ (𝐹𝑠)) = (𝑁 ∩ (𝐹𝑆)))
65neeq1d 2998 . . . . . . 7 (𝑠 = 𝑆 → ((𝑁 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹𝑆)) ≠ ∅))
73, 6rspc2v 3633 . . . . . 6 ((𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐿) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹𝑆)) ≠ ∅))
87ex 412 . . . . 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 481 . . 3 ((𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)))
111, 10biimtrdi 253 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆𝐿 → (𝑁 ∩ (𝐹𝑆)) ≠ ∅))))
12113imp2 1348 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 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  cin 3962  c0 4339  {csn 4631  cima 5692  wf 6559  cfv 6563  (class class class)co 7431  TopOnctopon 22932  neicnei 23121  Filcfil 23869   fClusf cfcf 23961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-fil 23870  df-fm 23962  df-fcls 23965  df-fcf 23966
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator