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Mirrors > Home > MPE Home > Th. List > fcfelbas | Structured version Visualization version GIF version |
Description: A cluster point of a function is in the base set of the topology. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
Ref | Expression |
---|---|
fcfelbas | β’ (((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ π΄ β ((π½ fClusf πΏ)βπΉ)) β π΄ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcfval 23407 | . . . . 5 β’ ((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β ((π½ fClusf πΏ)βπΉ) = (π½ fClus ((π FilMap πΉ)βπΏ))) | |
2 | 1 | eleq2d 2820 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β (π΄ β ((π½ fClusf πΏ)βπΉ) β π΄ β (π½ fClus ((π FilMap πΉ)βπΏ)))) |
3 | eqid 2733 | . . . . 5 β’ βͺ π½ = βͺ π½ | |
4 | 3 | fclselbas 23390 | . . . 4 β’ (π΄ β (π½ fClus ((π FilMap πΉ)βπΏ)) β π΄ β βͺ π½) |
5 | 2, 4 | syl6bi 253 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β (π΄ β ((π½ fClusf πΏ)βπΉ) β π΄ β βͺ π½)) |
6 | 5 | imp 408 | . 2 β’ (((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ π΄ β ((π½ fClusf πΏ)βπΉ)) β π΄ β βͺ π½) |
7 | simpl1 1192 | . . 3 β’ (((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ π΄ β ((π½ fClusf πΏ)βπΉ)) β π½ β (TopOnβπ)) | |
8 | toponuni 22286 | . . 3 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
9 | 7, 8 | syl 17 | . 2 β’ (((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ π΄ β ((π½ fClusf πΏ)βπΉ)) β π = βͺ π½) |
10 | 6, 9 | eleqtrrd 2837 | 1 β’ (((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ π΄ β ((π½ fClusf πΏ)βπΉ)) β π΄ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βͺ cuni 4869 βΆwf 6496 βcfv 6500 (class class class)co 7361 TopOnctopon 22282 Filcfil 23219 FilMap cfm 23307 fClus cfcls 23310 fClusf cfcf 23311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-map 8773 df-fbas 20816 df-top 22266 df-topon 22283 df-cld 22393 df-ntr 22394 df-cls 22395 df-fil 23220 df-fcls 23315 df-fcf 23316 |
This theorem is referenced by: (None) |
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