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Theorem uffcfflf 23898
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 22783 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
2 fmufil 23818 . . . 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 23890 . . 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 23763 . . 3 (𝐿 ∈ (UFilβ€˜π‘Œ) β†’ 𝐿 ∈ (Filβ€˜π‘Œ))
7 fcfval 23892 . . 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 23849 . . 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 2776 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 6533  β€˜cfv 6537  (class class class)co 7405  TopOnctopon 22767  Filcfil 23704  UFilcufil 23758   FilMap cfm 23792   fLim cflim 23793   fLimf cflf 23794   fClus cfcls 23795   fClusf cfcf 23796
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-fin 8945  df-fi 9408  df-fbas 21237  df-fg 21238  df-top 22751  df-topon 22768  df-cld 22878  df-ntr 22879  df-cls 22880  df-nei 22957  df-fil 23705  df-ufil 23760  df-fm 23797  df-flim 23798  df-flf 23799  df-fcls 23800  df-fcf 23801
This theorem is referenced by: (None)
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