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Mirrors > Home > MPE Home > Th. List > fclselbas | Structured version Visualization version GIF version |
Description: A cluster point is in the base set. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
fclselbas.1 | β’ π = βͺ π½ |
Ref | Expression |
---|---|
fclselbas | β’ (π΄ β (π½ fClus πΉ) β π΄ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fclselbas.1 | . . . . . 6 β’ π = βͺ π½ | |
2 | 1 | fclsfil 23835 | . . . . 5 β’ (π΄ β (π½ fClus πΉ) β πΉ β (Filβπ)) |
3 | fclstopon 23837 | . . . . 5 β’ (π΄ β (π½ fClus πΉ) β (π½ β (TopOnβπ) β πΉ β (Filβπ))) | |
4 | 2, 3 | mpbird 257 | . . . 4 β’ (π΄ β (π½ fClus πΉ) β π½ β (TopOnβπ)) |
5 | fclsopn 23839 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β (π΄ β (π½ fClus πΉ) β (π΄ β π β§ βπ β π½ (π΄ β π β βπ β πΉ (π β© π ) β β )))) | |
6 | 4, 2, 5 | syl2anc 583 | . . 3 β’ (π΄ β (π½ fClus πΉ) β (π΄ β (π½ fClus πΉ) β (π΄ β π β§ βπ β π½ (π΄ β π β βπ β πΉ (π β© π ) β β )))) |
7 | 6 | ibi 267 | . 2 β’ (π΄ β (π½ fClus πΉ) β (π΄ β π β§ βπ β π½ (π΄ β π β βπ β πΉ (π β© π ) β β ))) |
8 | 7 | simpld 494 | 1 β’ (π΄ β (π½ fClus πΉ) β π΄ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2932 βwral 3053 β© cin 3939 β c0 4314 βͺ cuni 4899 βcfv 6533 (class class class)co 7401 TopOnctopon 22733 Filcfil 23670 fClus cfcls 23761 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-fbas 21224 df-top 22717 df-topon 22734 df-cld 22844 df-ntr 22845 df-cls 22846 df-fil 23671 df-fcls 23766 |
This theorem is referenced by: fclsneii 23842 fclsnei 23844 fclsfnflim 23852 flimfnfcls 23853 fcfelbas 23861 cnfcf 23867 cfilfcls 25123 |
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