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Theorem flffbas 23499
Description: Limit points of a function can be defined using filter bases. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
Hypothesis
Ref Expression
flffbas.l 𝐿 = (π‘ŒfilGen𝐡)
Assertion
Ref Expression
flffbas ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ))))
Distinct variable groups:   π‘œ,𝑠,𝐴   𝐡,π‘œ,𝑠   π‘œ,𝐹,𝑠   π‘œ,𝐽,𝑠   π‘œ,𝐿,𝑠   π‘œ,𝑋,𝑠   π‘œ,π‘Œ,𝑠
Proof of Theorem flffbas
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 flffbas.l . . . 4 𝐿 = (π‘ŒfilGen𝐡)
2 fgcl 23382 . . . 4 (𝐡 ∈ (fBasβ€˜π‘Œ) β†’ (π‘ŒfilGen𝐡) ∈ (Filβ€˜π‘Œ))
31, 2eqeltrid 2838 . . 3 (𝐡 ∈ (fBasβ€˜π‘Œ) β†’ 𝐿 ∈ (Filβ€˜π‘Œ))
4 isflf 23497 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆƒπ‘‘ ∈ 𝐿 (𝐹 β€œ 𝑑) βŠ† π‘œ))))
53, 4syl3an2 1165 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆƒπ‘‘ ∈ 𝐿 (𝐹 β€œ 𝑑) βŠ† π‘œ))))
61eleq2i 2826 . . . . . . . 8 (𝑑 ∈ 𝐿 ↔ 𝑑 ∈ (π‘ŒfilGen𝐡))
7 elfg 23375 . . . . . . . . . . 11 (𝐡 ∈ (fBasβ€˜π‘Œ) β†’ (𝑑 ∈ (π‘ŒfilGen𝐡) ↔ (𝑑 βŠ† π‘Œ ∧ βˆƒπ‘  ∈ 𝐡 𝑠 βŠ† 𝑑)))
873ad2ant2 1135 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝑑 ∈ (π‘ŒfilGen𝐡) ↔ (𝑑 βŠ† π‘Œ ∧ βˆƒπ‘  ∈ 𝐡 𝑠 βŠ† 𝑑)))
9 sstr2 3990 . . . . . . . . . . . . . . . 16 ((𝐹 β€œ 𝑠) βŠ† (𝐹 β€œ 𝑑) β†’ ((𝐹 β€œ 𝑑) βŠ† π‘œ β†’ (𝐹 β€œ 𝑠) βŠ† π‘œ))
10 imass2 6102 . . . . . . . . . . . . . . . 16 (𝑠 βŠ† 𝑑 β†’ (𝐹 β€œ 𝑠) βŠ† (𝐹 β€œ 𝑑))
119, 10syl11 33 . . . . . . . . . . . . . . 15 ((𝐹 β€œ 𝑑) βŠ† π‘œ β†’ (𝑠 βŠ† 𝑑 β†’ (𝐹 β€œ 𝑠) βŠ† π‘œ))
1211adantl 483 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐹 β€œ 𝑑) βŠ† π‘œ) β†’ (𝑠 βŠ† 𝑑 β†’ (𝐹 β€œ 𝑠) βŠ† π‘œ))
1312reximdv 3171 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐹 β€œ 𝑑) βŠ† π‘œ) β†’ (βˆƒπ‘  ∈ 𝐡 𝑠 βŠ† 𝑑 β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ))
1413ex 414 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐹 β€œ 𝑑) βŠ† π‘œ β†’ (βˆƒπ‘  ∈ 𝐡 𝑠 βŠ† 𝑑 β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ)))
1514com23 86 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (βˆƒπ‘  ∈ 𝐡 𝑠 βŠ† 𝑑 β†’ ((𝐹 β€œ 𝑑) βŠ† π‘œ β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ)))
1615adantld 492 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝑑 βŠ† π‘Œ ∧ βˆƒπ‘  ∈ 𝐡 𝑠 βŠ† 𝑑) β†’ ((𝐹 β€œ 𝑑) βŠ† π‘œ β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ)))
178, 16sylbid 239 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝑑 ∈ (π‘ŒfilGen𝐡) β†’ ((𝐹 β€œ 𝑑) βŠ† π‘œ β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ)))
1817adantr 482 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝑑 ∈ (π‘ŒfilGen𝐡) β†’ ((𝐹 β€œ 𝑑) βŠ† π‘œ β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ)))
196, 18biimtrid 241 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝑑 ∈ 𝐿 β†’ ((𝐹 β€œ 𝑑) βŠ† π‘œ β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ)))
2019rexlimdv 3154 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (βˆƒπ‘‘ ∈ 𝐿 (𝐹 β€œ 𝑑) βŠ† π‘œ β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ))
21 ssfg 23376 . . . . . . . . . . . 12 (𝐡 ∈ (fBasβ€˜π‘Œ) β†’ 𝐡 βŠ† (π‘ŒfilGen𝐡))
2221, 1sseqtrrdi 4034 . . . . . . . . . . 11 (𝐡 ∈ (fBasβ€˜π‘Œ) β†’ 𝐡 βŠ† 𝐿)
2322sselda 3983 . . . . . . . . . 10 ((𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝑠 ∈ 𝐡) β†’ 𝑠 ∈ 𝐿)
24233ad2antl2 1187 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝑠 ∈ 𝐡) β†’ 𝑠 ∈ 𝐿)
2524ad2ant2r 746 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ 𝑋) ∧ (𝑠 ∈ 𝐡 ∧ (𝐹 β€œ 𝑠) βŠ† π‘œ)) β†’ 𝑠 ∈ 𝐿)
26 simprr 772 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ 𝑋) ∧ (𝑠 ∈ 𝐡 ∧ (𝐹 β€œ 𝑠) βŠ† π‘œ)) β†’ (𝐹 β€œ 𝑠) βŠ† π‘œ)
27 imaeq2 6056 . . . . . . . . . 10 (𝑑 = 𝑠 β†’ (𝐹 β€œ 𝑑) = (𝐹 β€œ 𝑠))
2827sseq1d 4014 . . . . . . . . 9 (𝑑 = 𝑠 β†’ ((𝐹 β€œ 𝑑) βŠ† π‘œ ↔ (𝐹 β€œ 𝑠) βŠ† π‘œ))
2928rspcev 3613 . . . . . . . 8 ((𝑠 ∈ 𝐿 ∧ (𝐹 β€œ 𝑠) βŠ† π‘œ) β†’ βˆƒπ‘‘ ∈ 𝐿 (𝐹 β€œ 𝑑) βŠ† π‘œ)
3025, 26, 29syl2anc 585 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ 𝑋) ∧ (𝑠 ∈ 𝐡 ∧ (𝐹 β€œ 𝑠) βŠ† π‘œ)) β†’ βˆƒπ‘‘ ∈ 𝐿 (𝐹 β€œ 𝑑) βŠ† π‘œ)
3130rexlimdvaa 3157 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ β†’ βˆƒπ‘‘ ∈ 𝐿 (𝐹 β€œ 𝑑) βŠ† π‘œ))
3220, 31impbid 211 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (βˆƒπ‘‘ ∈ 𝐿 (𝐹 β€œ 𝑑) βŠ† π‘œ ↔ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ))
3332imbi2d 341 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ 𝑋) β†’ ((𝐴 ∈ π‘œ β†’ βˆƒπ‘‘ ∈ 𝐿 (𝐹 β€œ 𝑑) βŠ† π‘œ) ↔ (𝐴 ∈ π‘œ β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ)))
3433ralbidv 3178 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆƒπ‘‘ ∈ 𝐿 (𝐹 β€œ 𝑑) βŠ† π‘œ) ↔ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ)))
3534pm5.32da 580 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐴 ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆƒπ‘‘ ∈ 𝐿 (𝐹 β€œ 𝑑) βŠ† π‘œ)) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ))))
365, 35bitrd 279 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆƒπ‘  ∈ 𝐡 (𝐹 β€œ 𝑠) βŠ† π‘œ))))
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   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  fBascfbas 20932  filGencfg 20933  TopOnctopon 22412  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-ntr 22524  df-nei 22602  df-fil 23350  df-fm 23442  df-flim 23443  df-flf 23444
This theorem is referenced by:  lmflf  23509  eltsms  23637
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