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Mathbox for Jeff Hankins |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tailf | Structured version Visualization version GIF version |
Description: The tail function of a directed set sends its elements to its subsets. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
Ref | Expression |
---|---|
tailf.1 | β’ π = dom π· |
Ref | Expression |
---|---|
tailf | β’ (π· β DirRel β (tailβπ·):πβΆπ« π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imassrn 6071 | . . . . . . 7 β’ (π· β {π₯}) β ran π· | |
2 | ssun2 4174 | . . . . . . . 8 β’ ran π· β (dom π· βͺ ran π·) | |
3 | dmrnssfld 5970 | . . . . . . . 8 β’ (dom π· βͺ ran π·) β βͺ βͺ π· | |
4 | 2, 3 | sstri 3992 | . . . . . . 7 β’ ran π· β βͺ βͺ π· |
5 | 1, 4 | sstri 3992 | . . . . . 6 β’ (π· β {π₯}) β βͺ βͺ π· |
6 | tailf.1 | . . . . . . 7 β’ π = dom π· | |
7 | dirdm 18559 | . . . . . . 7 β’ (π· β DirRel β dom π· = βͺ βͺ π·) | |
8 | 6, 7 | eqtr2id 2783 | . . . . . 6 β’ (π· β DirRel β βͺ βͺ π· = π) |
9 | 5, 8 | sseqtrid 4035 | . . . . 5 β’ (π· β DirRel β (π· β {π₯}) β π) |
10 | dmexg 7898 | . . . . . . 7 β’ (π· β DirRel β dom π· β V) | |
11 | 6, 10 | eqeltrid 2835 | . . . . . 6 β’ (π· β DirRel β π β V) |
12 | elpw2g 5345 | . . . . . 6 β’ (π β V β ((π· β {π₯}) β π« π β (π· β {π₯}) β π)) | |
13 | 11, 12 | syl 17 | . . . . 5 β’ (π· β DirRel β ((π· β {π₯}) β π« π β (π· β {π₯}) β π)) |
14 | 9, 13 | mpbird 256 | . . . 4 β’ (π· β DirRel β (π· β {π₯}) β π« π) |
15 | 14 | ralrimivw 3148 | . . 3 β’ (π· β DirRel β βπ₯ β π (π· β {π₯}) β π« π) |
16 | eqid 2730 | . . . 4 β’ (π₯ β π β¦ (π· β {π₯})) = (π₯ β π β¦ (π· β {π₯})) | |
17 | 16 | fmpt 7112 | . . 3 β’ (βπ₯ β π (π· β {π₯}) β π« π β (π₯ β π β¦ (π· β {π₯})):πβΆπ« π) |
18 | 15, 17 | sylib 217 | . 2 β’ (π· β DirRel β (π₯ β π β¦ (π· β {π₯})):πβΆπ« π) |
19 | 6 | tailfval 35562 | . . 3 β’ (π· β DirRel β (tailβπ·) = (π₯ β π β¦ (π· β {π₯}))) |
20 | 19 | feq1d 6703 | . 2 β’ (π· β DirRel β ((tailβπ·):πβΆπ« π β (π₯ β π β¦ (π· β {π₯})):πβΆπ« π)) |
21 | 18, 20 | mpbird 256 | 1 β’ (π· β DirRel β (tailβπ·):πβΆπ« π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1539 β wcel 2104 βwral 3059 Vcvv 3472 βͺ cun 3947 β wss 3949 π« cpw 4603 {csn 4629 βͺ cuni 4909 β¦ cmpt 5232 dom cdm 5677 ran crn 5678 β cima 5680 βΆwf 6540 βcfv 6544 DirRelcdir 18553 tailctail 18554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 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-dir 18555 df-tail 18556 |
This theorem is referenced by: tailfb 35567 filnetlem4 35571 |
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