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Theorem uffcfflf 23963
Description: If the domain filter is an ultrafilter, the cluster points of the function are the limit points. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
uffcfflf ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (UFilβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐽 fClusf 𝐿)β€˜πΉ) = ((𝐽 fLimf 𝐿)β€˜πΉ))
Proof of Theorem uffcfflf
StepHypRef Expression
1 toponmax 22848 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
2 fmufil 23883 . . . 4 ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (UFilβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝑋 FilMap 𝐹)β€˜πΏ) ∈ (UFilβ€˜π‘‹))
31, 2syl3an1 1160 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (UFilβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝑋 FilMap 𝐹)β€˜πΏ) ∈ (UFilβ€˜π‘‹))
4 uffclsflim 23955 . . 3 (((𝑋 FilMap 𝐹)β€˜πΏ) ∈ (UFilβ€˜π‘‹) β†’ (𝐽 fClus ((𝑋 FilMap 𝐹)β€˜πΏ)) = (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
53, 4syl 17 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (UFilβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐽 fClus ((𝑋 FilMap 𝐹)β€˜πΏ)) = (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
6 ufilfil 23828 . . 3 (𝐿 ∈ (UFilβ€˜π‘Œ) β†’ 𝐿 ∈ (Filβ€˜π‘Œ))
7 fcfval 23957 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐽 fClusf 𝐿)β€˜πΉ) = (𝐽 fClus ((𝑋 FilMap 𝐹)β€˜πΏ)))
86, 7syl3an2 1161 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (UFilβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐽 fClusf 𝐿)β€˜πΉ) = (𝐽 fClus ((𝑋 FilMap 𝐹)β€˜πΏ)))
9 flfval 23914 . . 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)))
106, 9syl3an2 1161 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (UFilβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐽 fLimf 𝐿)β€˜πΉ) = (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
115, 8, 103eqtr4d 2778 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (UFilβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐽 fClusf 𝐿)β€˜πΉ) = ((𝐽 fLimf 𝐿)β€˜πΉ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  TopOnctopon 22832  Filcfil 23769  UFilcufil 23823   FilMap cfm 23857   fLim cflim 23858   fLimf cflf 23859   fClus cfcls 23860   fClusf cfcf 23861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1o 8493  df-er 8731  df-map 8853  df-en 8971  df-fin 8974  df-fi 9442  df-fbas 21283  df-fg 21284  df-top 22816  df-topon 22833  df-cld 22943  df-ntr 22944  df-cls 22945  df-nei 23022  df-fil 23770  df-ufil 23825  df-fm 23862  df-flim 23863  df-flf 23864  df-fcls 23865  df-fcf 23866
This theorem is referenced by: (None)
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