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Mirrors > Home > MPE Home > Th. List > fcfneii | Structured version Visualization version GIF version |
Description: A neighborhood of a cluster point of a function contains a function value from every tail. (Contributed by Jeff Hankins, 27-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
Ref | Expression |
---|---|
fcfneii | β’ (((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ (π΄ β ((π½ fClusf πΏ)βπΉ) β§ π β ((neiβπ½)β{π΄}) β§ π β πΏ)) β (π β© (πΉ β π)) β β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcfnei 23889 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β (π΄ β ((π½ fClusf πΏ)βπΉ) β (π΄ β π β§ βπ β ((neiβπ½)β{π΄})βπ β πΏ (π β© (πΉ β π )) β β ))) | |
2 | ineq1 4200 | . . . . . . . 8 β’ (π = π β (π β© (πΉ β π )) = (π β© (πΉ β π ))) | |
3 | 2 | neeq1d 2994 | . . . . . . 7 β’ (π = π β ((π β© (πΉ β π )) β β β (π β© (πΉ β π )) β β )) |
4 | imaeq2 6048 | . . . . . . . . 9 β’ (π = π β (πΉ β π ) = (πΉ β π)) | |
5 | 4 | ineq2d 4207 | . . . . . . . 8 β’ (π = π β (π β© (πΉ β π )) = (π β© (πΉ β π))) |
6 | 5 | neeq1d 2994 | . . . . . . 7 β’ (π = π β ((π β© (πΉ β π )) β β β (π β© (πΉ β π)) β β )) |
7 | 3, 6 | rspc2v 3617 | . . . . . 6 β’ ((π β ((neiβπ½)β{π΄}) β§ π β πΏ) β (βπ β ((neiβπ½)β{π΄})βπ β πΏ (π β© (πΉ β π )) β β β (π β© (πΉ β π)) β β )) |
8 | 7 | ex 412 | . . . . 5 β’ (π β ((neiβπ½)β{π΄}) β (π β πΏ β (βπ β ((neiβπ½)β{π΄})βπ β πΏ (π β© (πΉ β π )) β β β (π β© (πΉ β π)) β β ))) |
9 | 8 | com3r 87 | . . . 4 β’ (βπ β ((neiβπ½)β{π΄})βπ β πΏ (π β© (πΉ β π )) β β β (π β ((neiβπ½)β{π΄}) β (π β πΏ β (π β© (πΉ β π)) β β ))) |
10 | 9 | adantl 481 | . . 3 β’ ((π΄ β π β§ βπ β ((neiβπ½)β{π΄})βπ β πΏ (π β© (πΉ β π )) β β ) β (π β ((neiβπ½)β{π΄}) β (π β πΏ β (π β© (πΉ β π)) β β ))) |
11 | 1, 10 | biimtrdi 252 | . 2 β’ ((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β (π΄ β ((π½ fClusf πΏ)βπΉ) β (π β ((neiβπ½)β{π΄}) β (π β πΏ β (π β© (πΉ β π)) β β )))) |
12 | 11 | 3imp2 1346 | 1 β’ (((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ (π΄ β ((π½ fClusf πΏ)βπΉ) β§ π β ((neiβπ½)β{π΄}) β§ π β πΏ)) β (π β© (πΉ β π)) β β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 β© cin 3942 β c0 4317 {csn 4623 β cima 5672 βΆwf 6532 βcfv 6536 (class class class)co 7404 TopOnctopon 22762 neicnei 22951 Filcfil 23699 fClusf cfcf 23791 |
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 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-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-id 5567 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-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8821 df-fbas 21232 df-fg 21233 df-top 22746 df-topon 22763 df-cld 22873 df-ntr 22874 df-cls 22875 df-nei 22952 df-fil 23700 df-fm 23792 df-fcls 23795 df-fcf 23796 |
This theorem is referenced by: (None) |
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