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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 23688 | . . . . 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 498 | . . . 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 1143 | . . 3 β’ (π΄ β (π½ fLim πΉ) β πΉ β βͺ ran Fil) |
5 | filunirn 23606 | . . 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 1144 | . . . . 5 β’ (π΄ β (π½ fLim πΉ) β πΉ β π« π) |
8 | sspwuni 5103 | . . . . 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 23690 | . . . . . 6 β’ (π΄ β (π½ fLim πΉ) β ((neiβπ½)β{π΄}) β πΉ) | |
11 | flimtop 23689 | . . . . . . 7 β’ (π΄ β (π½ fLim πΉ) β π½ β Top) | |
12 | 1 | topopn 22628 | . . . . . . . 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 23692 | . . . . . . 7 β’ (π΄ β (π½ fLim πΉ) β π΄ β π) |
15 | opnneip 22843 | . . . . . . 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 1371 | . . . . . 6 β’ (π΄ β (π½ fLim πΉ) β π β ((neiβπ½)β{π΄})) |
17 | 10, 16 | sseldd 3983 | . . . . 5 β’ (π΄ β (π½ fLim πΉ) β π β πΉ) |
18 | elssuni 4941 | . . . . 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 3999 | . . 3 β’ (π΄ β (π½ fLim πΉ) β βͺ πΉ = π) |
21 | 20 | fveq2d 6895 | . 2 β’ (π΄ β (π½ fLim πΉ) β (Filββͺ πΉ) = (Filβπ)) |
22 | 6, 21 | eleqtrd 2835 | 1 β’ (π΄ β (π½ fLim πΉ) β πΉ β (Filβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wss 3948 π« cpw 4602 {csn 4628 βͺ cuni 4908 ran crn 5677 βcfv 6543 (class class class)co 7411 Topctop 22615 neicnei 22821 Filcfil 23569 fLim cflim 23658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-fbas 21141 df-top 22616 df-nei 22822 df-fil 23570 df-flim 23663 |
This theorem is referenced by: flimtopon 23694 flimss1 23697 flimclsi 23702 hausflimlem 23703 flimsncls 23710 cnpflfi 23723 flimfcls 23750 flimcfil 25055 |
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