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Mirrors > Home > MPE Home > Th. List > uffcfflf | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
uffcfflf | β’ ((π½ β (TopOnβπ) β§ πΏ β (UFilβπ) β§ πΉ:πβΆπ) β ((π½ fClusf πΏ)βπΉ) = ((π½ fLimf πΏ)βπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponmax 22419 | . . . 4 β’ (π½ β (TopOnβπ) β π β π½) | |
2 | fmufil 23454 | . . . 4 β’ ((π β π½ β§ πΏ β (UFilβπ) β§ πΉ:πβΆπ) β ((π FilMap πΉ)βπΏ) β (UFilβπ)) | |
3 | 1, 2 | syl3an1 1163 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΏ β (UFilβπ) β§ πΉ:πβΆπ) β ((π FilMap πΉ)βπΏ) β (UFilβπ)) |
4 | uffclsflim 23526 | . . 3 β’ (((π FilMap πΉ)βπΏ) β (UFilβπ) β (π½ fClus ((π FilMap πΉ)βπΏ)) = (π½ fLim ((π FilMap πΉ)βπΏ))) | |
5 | 3, 4 | syl 17 | . 2 β’ ((π½ β (TopOnβπ) β§ πΏ β (UFilβπ) β§ πΉ:πβΆπ) β (π½ fClus ((π FilMap πΉ)βπΏ)) = (π½ fLim ((π FilMap πΉ)βπΏ))) |
6 | ufilfil 23399 | . . 3 β’ (πΏ β (UFilβπ) β πΏ β (Filβπ)) | |
7 | fcfval 23528 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β ((π½ fClusf πΏ)βπΉ) = (π½ fClus ((π FilMap πΉ)βπΏ))) | |
8 | 6, 7 | syl3an2 1164 | . 2 β’ ((π½ β (TopOnβπ) β§ πΏ β (UFilβπ) β§ πΉ:πβΆπ) β ((π½ fClusf πΏ)βπΉ) = (π½ fClus ((π FilMap πΉ)βπΏ))) |
9 | flfval 23485 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β ((π½ fLimf πΏ)βπΉ) = (π½ fLim ((π FilMap πΉ)βπΏ))) | |
10 | 6, 9 | syl3an2 1164 | . 2 β’ ((π½ β (TopOnβπ) β§ πΏ β (UFilβπ) β§ πΉ:πβΆπ) β ((π½ fLimf πΏ)βπΉ) = (π½ fLim ((π FilMap πΉ)βπΏ))) |
11 | 5, 8, 10 | 3eqtr4d 2782 | 1 β’ ((π½ β (TopOnβπ) β§ πΏ β (UFilβπ) β§ πΉ:πβΆπ) β ((π½ fClusf πΏ)βπΉ) = ((π½ fLimf πΏ)βπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1087 = wceq 1541 β wcel 2106 βΆwf 6536 βcfv 6540 (class class class)co 7405 TopOnctopon 22403 Filcfil 23340 UFilcufil 23394 FilMap cfm 23428 fLim cflim 23429 fLimf cflf 23430 fClus cfcls 23431 fClusf cfcf 23432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-fin 8939 df-fi 9402 df-fbas 20933 df-fg 20934 df-top 22387 df-topon 22404 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-fil 23341 df-ufil 23396 df-fm 23433 df-flim 23434 df-flf 23435 df-fcls 23436 df-fcf 23437 |
This theorem is referenced by: (None) |
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