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Theorem flfneii 23359
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 = βˆͺ 𝐽
Assertion
Ref Expression
flfneii (((𝐽 ∈ Top ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑁)
Distinct variable groups:   𝐹,𝑠   𝐽,𝑠   𝐿,𝑠   𝑁,𝑠   𝑋,𝑠   π‘Œ,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 = βˆͺ 𝐽
21toptopon 22282 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
3 flfnei 23358 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛)))
42, 3syl3an1b 1404 . . . 4 ((𝐽 ∈ Top ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛)))
54simplbda 501 . . 3 (((𝐽 ∈ Top ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ)) β†’ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛)
653adant3 1133 . 2 (((𝐽 ∈ Top ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛)
7 sseq2 3975 . . . . 5 (𝑛 = 𝑁 β†’ ((𝐹 β€œ 𝑠) βŠ† 𝑛 ↔ (𝐹 β€œ 𝑠) βŠ† 𝑁))
87rexbidv 3176 . . . 4 (𝑛 = 𝑁 β†’ (βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛 ↔ βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑁))
98rspcv 3580 . . 3 (𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴}) β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛 β†’ βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑁))
1093ad2ant3 1136 . 2 (((𝐽 ∈ Top ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛 β†’ βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑁))
116, 10mpd 15 1 (((𝐽 ∈ Top ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑁)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074   βŠ† wss 3915  {csn 4591  βˆͺ cuni 4870   β€œ cima 5641  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  Topctop 22258  TopOnctopon 22275  neicnei 22464  Filcfil 23212   fLimf cflf 23302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774  df-fbas 20809  df-fg 20810  df-top 22259  df-topon 22276  df-nei 22465  df-fil 23213  df-fm 23305  df-flim 23306  df-flf 23307
This theorem is referenced by: (None)
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