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Mirrors > Home > MPE Home > Th. List > flfneii | Structured version Visualization version GIF version |
Description: A neighborhood of a limit point of a function contains the image of a filter element. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
Ref | Expression |
---|---|
flfneii.x | β’ π = βͺ π½ |
Ref | Expression |
---|---|
flfneii | β’ (((π½ β Top β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ π΄ β ((π½ fLimf πΏ)βπΉ) β§ π β ((neiβπ½)β{π΄})) β βπ β πΏ (πΉ β π ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flfneii.x | . . . . . 6 β’ π = βͺ π½ | |
2 | 1 | toptopon 22282 | . . . . 5 β’ (π½ β Top β π½ β (TopOnβπ)) |
3 | flfnei 23358 | . . . . 5 β’ ((π½ β (TopOnβπ) β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β (π΄ β ((π½ fLimf πΏ)βπΉ) β (π΄ β π β§ βπ β ((neiβπ½)β{π΄})βπ β πΏ (πΉ β π ) β π))) | |
4 | 2, 3 | syl3an1b 1404 | . . . 4 β’ ((π½ β Top β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β (π΄ β ((π½ fLimf πΏ)βπΉ) β (π΄ β π β§ βπ β ((neiβπ½)β{π΄})βπ β πΏ (πΉ β π ) β π))) |
5 | 4 | simplbda 501 | . . 3 β’ (((π½ β Top β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ π΄ β ((π½ fLimf πΏ)βπΉ)) β βπ β ((neiβπ½)β{π΄})βπ β πΏ (πΉ β π ) β π) |
6 | 5 | 3adant3 1133 | . 2 β’ (((π½ β Top β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ π΄ β ((π½ fLimf πΏ)βπΉ) β§ π β ((neiβπ½)β{π΄})) β βπ β ((neiβπ½)β{π΄})βπ β πΏ (πΉ β π ) β π) |
7 | sseq2 3975 | . . . . 5 β’ (π = π β ((πΉ β π ) β π β (πΉ β π ) β π)) | |
8 | 7 | rexbidv 3176 | . . . 4 β’ (π = π β (βπ β πΏ (πΉ β π ) β π β βπ β πΏ (πΉ β π ) β π)) |
9 | 8 | rspcv 3580 | . . 3 β’ (π β ((neiβπ½)β{π΄}) β (βπ β ((neiβπ½)β{π΄})βπ β πΏ (πΉ β π ) β π β βπ β πΏ (πΉ β π ) β π)) |
10 | 9 | 3ad2ant3 1136 | . 2 β’ (((π½ β Top β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ π΄ β ((π½ fLimf πΏ)βπΉ) β§ π β ((neiβπ½)β{π΄})) β (βπ β ((neiβπ½)β{π΄})βπ β πΏ (πΉ β π ) β π β βπ β πΏ (πΉ β π ) β π)) |
11 | 6, 10 | mpd 15 | 1 β’ (((π½ β Top β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ π΄ β ((π½ fLimf πΏ)βπΉ) β§ π β ((neiβπ½)β{π΄})) β βπ β πΏ (πΉ β π ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3065 βwrex 3074 β wss 3915 {csn 4591 βͺ cuni 4870 β cima 5641 βΆwf 6497 βcfv 6501 (class class class)co 7362 Topctop 22258 TopOnctopon 22275 neicnei 22464 Filcfil 23212 fLimf cflf 23302 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-map 8774 df-fbas 20809 df-fg 20810 df-top 22259 df-topon 22276 df-nei 22465 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 |
This theorem is referenced by: (None) |
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