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Mirrors > Home > MPE Home > Th. List > Mathboxes > filnet | Structured version Visualization version GIF version |
Description: A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.) |
Ref | Expression |
---|---|
filnet | β’ (πΉ β (Filβπ) β βπ β DirRel βπ(π:dom πβΆπ β§ πΉ = ((π FilMap π)βran (tailβπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . 2 β’ βͺ π β πΉ ({π} Γ π) = βͺ π β πΉ ({π} Γ π) | |
2 | eqid 2733 | . 2 β’ {β¨π₯, π¦β© β£ ((π₯ β βͺ π β πΉ ({π} Γ π) β§ π¦ β βͺ π β πΉ ({π} Γ π)) β§ (1st βπ¦) β (1st βπ₯))} = {β¨π₯, π¦β© β£ ((π₯ β βͺ π β πΉ ({π} Γ π) β§ π¦ β βͺ π β πΉ ({π} Γ π)) β§ (1st βπ¦) β (1st βπ₯))} | |
3 | 1, 2 | filnetlem4 35266 | 1 β’ (πΉ β (Filβπ) β βπ β DirRel βπ(π:dom πβΆπ β§ πΉ = ((π FilMap π)βran (tailβπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 βwex 1782 β wcel 2107 βwrex 3071 β wss 3949 {csn 4629 βͺ ciun 4998 {copab 5211 Γ cxp 5675 dom cdm 5677 ran crn 5678 βΆwf 6540 βcfv 6544 (class class class)co 7409 1st c1st 7973 DirRelcdir 18547 tailctail 18548 Filcfil 23349 FilMap cfm 23437 |
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 ax-un 7725 |
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-1st 7975 df-2nd 7976 df-dir 18549 df-tail 18550 df-fbas 20941 df-fg 20942 df-fil 23350 df-fm 23442 |
This theorem is referenced by: (None) |
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