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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 23959 | . . 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 4207 | . . . . . . . 8 β’ (π = π β (π β© (πΉ β π )) = (π β© (πΉ β π ))) | |
3 | 2 | neeq1d 2997 | . . . . . . 7 β’ (π = π β ((π β© (πΉ β π )) β β β (π β© (πΉ β π )) β β )) |
4 | imaeq2 6064 | . . . . . . . . 9 β’ (π = π β (πΉ β π ) = (πΉ β π)) | |
5 | 4 | ineq2d 4214 | . . . . . . . 8 β’ (π = π β (π β© (πΉ β π )) = (π β© (πΉ β π))) |
6 | 5 | neeq1d 2997 | . . . . . . 7 β’ (π = π β ((π β© (πΉ β π )) β β β (π β© (πΉ β π)) β β )) |
7 | 3, 6 | rspc2v 3622 | . . . . . 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 411 | . . . . 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 480 | . . 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2937 βwral 3058 β© cin 3948 β c0 4326 {csn 4632 β cima 5685 βΆwf 6549 βcfv 6553 (class class class)co 7426 TopOnctopon 22832 neicnei 23021 Filcfil 23769 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-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-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-id 5580 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-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-map 8853 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-fm 23862 df-fcls 23865 df-fcf 23866 |
This theorem is referenced by: (None) |
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