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Mirrors  >  Home  >  MPE Home  >  Th. List  >  flfnei Structured version   Visualization version   GIF version
Theorem flfnei 23717
Description: The property of being a limit point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
flfnei ((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)))
Distinct variable groups:   𝑛,𝑠,𝐹   𝐴,𝑛   𝑛,𝐽,𝑠   𝑛,𝐿,𝑠   𝑛,𝑋,𝑠   𝑛,π‘Œ,𝑠
Allowed substitution hint:   𝐴(𝑠)
Proof of Theorem flfnei
StepHypRef Expression
1 flfval 23716 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐽 fLimf 𝐿)β€˜πΉ) = (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
21eleq2d 2817 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ↔ 𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ))))
3 simp1 1134 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
4 toponmax 22650 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
543ad2ant1 1131 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ 𝑋 ∈ 𝐽)
6 filfbas 23574 . . . . 5 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ 𝐿 ∈ (fBasβ€˜π‘Œ))
763ad2ant2 1132 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ 𝐿 ∈ (fBasβ€˜π‘Œ))
8 simp3 1136 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ 𝐹:π‘ŒβŸΆπ‘‹)
9 fmfil 23670 . . . 4 ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝑋 FilMap 𝐹)β€˜πΏ) ∈ (Filβ€˜π‘‹))
105, 7, 8, 9syl3anc 1369 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝑋 FilMap 𝐹)β€˜πΏ) ∈ (Filβ€˜π‘‹))
11 elflim 23697 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ ((𝑋 FilMap 𝐹)β€˜πΏ) ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)) ↔ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† ((𝑋 FilMap 𝐹)β€˜πΏ))))
123, 10, 11syl2anc 582 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)) ↔ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† ((𝑋 FilMap 𝐹)β€˜πΏ))))
13 dfss3 3971 . . . 4 (((neiβ€˜π½)β€˜{𝐴}) βŠ† ((𝑋 FilMap 𝐹)β€˜πΏ) ↔ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})𝑛 ∈ ((𝑋 FilMap 𝐹)β€˜πΏ))
14 topontop 22637 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
15143ad2ant1 1131 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ 𝐽 ∈ Top)
16 eqid 2730 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
1716neii1 22832 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑛 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝑛 βŠ† βˆͺ 𝐽)
1815, 17sylan 578 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝑛 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝑛 βŠ† βˆͺ 𝐽)
19 toponuni 22638 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
20193ad2ant1 1131 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ 𝑋 = βˆͺ 𝐽)
2120adantr 479 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝑛 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝑋 = βˆͺ 𝐽)
2218, 21sseqtrrd 4024 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝑛 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝑛 βŠ† 𝑋)
23 elfm 23673 . . . . . . . 8 ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝑛 ∈ ((𝑋 FilMap 𝐹)β€˜πΏ) ↔ (𝑛 βŠ† 𝑋 ∧ βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛)))
245, 7, 8, 23syl3anc 1369 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝑛 ∈ ((𝑋 FilMap 𝐹)β€˜πΏ) ↔ (𝑛 βŠ† 𝑋 ∧ βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛)))
2524baibd 538 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝑛 βŠ† 𝑋) β†’ (𝑛 ∈ ((𝑋 FilMap 𝐹)β€˜πΏ) ↔ βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛))
2622, 25syldan 589 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝑛 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (𝑛 ∈ ((𝑋 FilMap 𝐹)β€˜πΏ) ↔ βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛))
2726ralbidva 3173 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})𝑛 ∈ ((𝑋 FilMap 𝐹)β€˜πΏ) ↔ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛))
2813, 27bitrid 282 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (((neiβ€˜π½)β€˜{𝐴}) βŠ† ((𝑋 FilMap 𝐹)β€˜πΏ) ↔ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛))
2928anbi2d 627 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† ((𝑋 FilMap 𝐹)β€˜πΏ)) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘  ∈ 𝐿 (𝐹 β€œ 𝑠) βŠ† 𝑛)))
302, 12, 293bitrd 304 1 ((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)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068   βŠ† wss 3949  {csn 4629  βˆͺ cuni 4909   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7413  fBascfbas 21134  Topctop 22617  TopOnctopon 22634  neicnei 22823  Filcfil 23571   FilMap cfm 23659   fLim cflim 23660   fLimf cflf 23661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 7416  df-oprab 7417  df-mpo 7418  df-map 8826  df-fbas 21143  df-fg 21144  df-top 22618  df-topon 22635  df-nei 22824  df-fil 23572  df-fm 23664  df-flim 23665  df-flf 23666
This theorem is referenced by:  flfneii  23718  cnextcn  23793  cnextfres1  23794
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