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Mirrors > Home > MPE Home > Th. List > flimfil | Structured version Visualization version GIF version |
Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
Ref | Expression |
---|---|
flimuni.1 | β’ π = βͺ π½ |
Ref | Expression |
---|---|
flimfil | β’ (π΄ β (π½ fLim πΉ) β πΉ β (Filβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flimuni.1 | . . . . . 6 β’ π = βͺ π½ | |
2 | 1 | elflim2 23468 | . . . . 5 β’ (π΄ β (π½ fLim πΉ) β ((π½ β Top β§ πΉ β βͺ ran Fil β§ πΉ β π« π) β§ (π΄ β π β§ ((neiβπ½)β{π΄}) β πΉ))) |
3 | 2 | simplbi 499 | . . . 4 β’ (π΄ β (π½ fLim πΉ) β (π½ β Top β§ πΉ β βͺ ran Fil β§ πΉ β π« π)) |
4 | 3 | simp2d 1144 | . . 3 β’ (π΄ β (π½ fLim πΉ) β πΉ β βͺ ran Fil) |
5 | filunirn 23386 | . . 3 β’ (πΉ β βͺ ran Fil β πΉ β (Filββͺ πΉ)) | |
6 | 4, 5 | sylib 217 | . 2 β’ (π΄ β (π½ fLim πΉ) β πΉ β (Filββͺ πΉ)) |
7 | 3 | simp3d 1145 | . . . . 5 β’ (π΄ β (π½ fLim πΉ) β πΉ β π« π) |
8 | sspwuni 5104 | . . . . 5 β’ (πΉ β π« π β βͺ πΉ β π) | |
9 | 7, 8 | sylib 217 | . . . 4 β’ (π΄ β (π½ fLim πΉ) β βͺ πΉ β π) |
10 | flimneiss 23470 | . . . . . 6 β’ (π΄ β (π½ fLim πΉ) β ((neiβπ½)β{π΄}) β πΉ) | |
11 | flimtop 23469 | . . . . . . 7 β’ (π΄ β (π½ fLim πΉ) β π½ β Top) | |
12 | 1 | topopn 22408 | . . . . . . . 8 β’ (π½ β Top β π β π½) |
13 | 11, 12 | syl 17 | . . . . . . 7 β’ (π΄ β (π½ fLim πΉ) β π β π½) |
14 | 1 | flimelbas 23472 | . . . . . . 7 β’ (π΄ β (π½ fLim πΉ) β π΄ β π) |
15 | opnneip 22623 | . . . . . . 7 β’ ((π½ β Top β§ π β π½ β§ π΄ β π) β π β ((neiβπ½)β{π΄})) | |
16 | 11, 13, 14, 15 | syl3anc 1372 | . . . . . 6 β’ (π΄ β (π½ fLim πΉ) β π β ((neiβπ½)β{π΄})) |
17 | 10, 16 | sseldd 3984 | . . . . 5 β’ (π΄ β (π½ fLim πΉ) β π β πΉ) |
18 | elssuni 4942 | . . . . 5 β’ (π β πΉ β π β βͺ πΉ) | |
19 | 17, 18 | syl 17 | . . . 4 β’ (π΄ β (π½ fLim πΉ) β π β βͺ πΉ) |
20 | 9, 19 | eqssd 4000 | . . 3 β’ (π΄ β (π½ fLim πΉ) β βͺ πΉ = π) |
21 | 20 | fveq2d 6896 | . 2 β’ (π΄ β (π½ fLim πΉ) β (Filββͺ πΉ) = (Filβπ)) |
22 | 6, 21 | eleqtrd 2836 | 1 β’ (π΄ β (π½ fLim πΉ) β πΉ β (Filβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wss 3949 π« cpw 4603 {csn 4629 βͺ cuni 4909 ran crn 5678 βcfv 6544 (class class class)co 7409 Topctop 22395 neicnei 22601 Filcfil 23349 fLim cflim 23438 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-fbas 20941 df-top 22396 df-nei 22602 df-fil 23350 df-flim 23443 |
This theorem is referenced by: flimtopon 23474 flimss1 23477 flimclsi 23482 hausflimlem 23483 flimsncls 23490 cnpflfi 23503 flimfcls 23530 flimcfil 24831 |
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