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Theorem flfneii 23496
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 22419 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
3 flfnei 23495 . . . . 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 4009 . . . . 5 (𝑛 = 𝑁 β†’ ((𝐹 β€œ 𝑠) βŠ† 𝑛 ↔ (𝐹 β€œ 𝑠) βŠ† 𝑁))
87rexbidv 3179 . . . 4 (𝑛 = 𝑁 β†’ (βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛 ↔ βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑁))
98rspcv 3609 . . 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 3062  βˆƒwrex 3071   βŠ† wss 3949  {csn 4629  βˆͺ cuni 4909   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Topctop 22395  TopOnctopon 22412  neicnei 22601  Filcfil 23349   fLimf cflf 23439
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-nei 22602  df-fil 23350  df-fm 23442  df-flim 23443  df-flf 23444
This theorem is referenced by: (None)
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