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Theorem flfneii 22319
Description: A neighborhood of a limit point of a function contains the image of a filter element. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flfneii.x 𝑋 = 𝐽
Assertion
Ref Expression
flfneii (((𝐽 ∈ Top ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑁)
Distinct variable groups:   𝐹,𝑠   𝐽,𝑠   𝐿,𝑠   𝑁,𝑠   𝑋,𝑠   𝑌,𝑠
Allowed substitution hint:   𝐴(𝑠)

Proof of Theorem flfneii
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 flfneii.x . . . . . 6 𝑋 = 𝐽
21toptopon 21244 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3 flfnei 22318 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛)))
42, 3syl3an1b 1384 . . . 4 ((𝐽 ∈ Top ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛)))
54simplbda 492 . . 3 (((𝐽 ∈ Top ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛)
653adant3 1113 . 2 (((𝐽 ∈ Top ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛)
7 sseq2 3876 . . . . 5 (𝑛 = 𝑁 → ((𝐹𝑠) ⊆ 𝑛 ↔ (𝐹𝑠) ⊆ 𝑁))
87rexbidv 3235 . . . 4 (𝑛 = 𝑁 → (∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛 ↔ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑁))
98rspcv 3524 . . 3 (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑁))
1093ad2ant3 1116 . 2 (((𝐽 ∈ Top ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑁))
116, 10mpd 15 1 (((𝐽 ∈ Top ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069   = wceq 1508  wcel 2051  wral 3081  wrex 3082  wss 3822  {csn 4435   cuni 4708  cima 5406  wf 6181  cfv 6185  (class class class)co 6974  Topctop 21220  TopOnctopon 21237  neicnei 21424  Filcfil 22172   fLimf cflf 22262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-map 8206  df-fbas 20259  df-fg 20260  df-top 21221  df-topon 21238  df-nei 21425  df-fil 22173  df-fm 22265  df-flim 22266  df-flf 22267
This theorem is referenced by: (None)
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