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Mirrors > Home > MPE Home > Th. List > disjxwwlkn | Structured version Visualization version GIF version |
Description: Sets of walks (as words) extended by an edge are disjunct if each set contains extensions of distinct walks. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 26-Oct-2022.) |
Ref | Expression |
---|---|
wwlksnextprop.x | β’ π = ((π + 1) WWalksN πΊ) |
wwlksnextprop.e | β’ πΈ = (EdgβπΊ) |
wwlksnextprop.y | β’ π = {π€ β (π WWalksN πΊ) β£ (π€β0) = π} |
Ref | Expression |
---|---|
disjxwwlkn | β’ Disj π¦ β π {π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . . . . . 6 β’ (((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ) β (π₯ prefix π) = π¦) | |
2 | 1 | a1i 11 | . . . . 5 β’ (π₯ β π β (((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ) β (π₯ prefix π) = π¦)) |
3 | 2 | ss2rabi 4022 | . . . 4 β’ {π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)} β {π₯ β π β£ (π₯ prefix π) = π¦} |
4 | wwlksnextprop.x | . . . . . 6 β’ π = ((π + 1) WWalksN πΊ) | |
5 | wwlkssswwlksn 28519 | . . . . . . 7 β’ ((π + 1) WWalksN πΊ) β (WWalksβπΊ) | |
6 | eqid 2736 | . . . . . . . 8 β’ (VtxβπΊ) = (VtxβπΊ) | |
7 | 6 | wwlkssswrd 28515 | . . . . . . 7 β’ (WWalksβπΊ) β Word (VtxβπΊ) |
8 | 5, 7 | sstri 3941 | . . . . . 6 β’ ((π + 1) WWalksN πΊ) β Word (VtxβπΊ) |
9 | 4, 8 | eqsstri 3966 | . . . . 5 β’ π β Word (VtxβπΊ) |
10 | rabss2 4023 | . . . . 5 β’ (π β Word (VtxβπΊ) β {π₯ β π β£ (π₯ prefix π) = π¦} β {π₯ β Word (VtxβπΊ) β£ (π₯ prefix π) = π¦}) | |
11 | 9, 10 | ax-mp 5 | . . . 4 β’ {π₯ β π β£ (π₯ prefix π) = π¦} β {π₯ β Word (VtxβπΊ) β£ (π₯ prefix π) = π¦} |
12 | 3, 11 | sstri 3941 | . . 3 β’ {π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)} β {π₯ β Word (VtxβπΊ) β£ (π₯ prefix π) = π¦} |
13 | 12 | rgenw 3065 | . 2 β’ βπ¦ β π {π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)} β {π₯ β Word (VtxβπΊ) β£ (π₯ prefix π) = π¦} |
14 | disjwrdpfx 14511 | . 2 β’ Disj π¦ β π {π₯ β Word (VtxβπΊ) β£ (π₯ prefix π) = π¦} | |
15 | disjss2 5060 | . 2 β’ (βπ¦ β π {π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)} β {π₯ β Word (VtxβπΊ) β£ (π₯ prefix π) = π¦} β (Disj π¦ β π {π₯ β Word (VtxβπΊ) β£ (π₯ prefix π) = π¦} β Disj π¦ β π {π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)})) | |
16 | 13, 14, 15 | mp2 9 | 1 β’ Disj π¦ β π {π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)} |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1086 = wceq 1540 β wcel 2105 βwral 3061 {crab 3403 β wss 3898 {cpr 4575 Disj wdisj 5057 βcfv 6479 (class class class)co 7337 0cc0 10972 1c1 10973 + caddc 10975 Word cword 14317 lastSclsw 14365 prefix cpfx 14481 Vtxcvtx 27655 Edgcedg 27706 WWalkscwwlks 28478 WWalksN cwwlksn 28479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-disj 5058 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-n0 12335 df-z 12421 df-uz 12684 df-fz 13341 df-fzo 13484 df-hash 14146 df-word 14318 df-wwlks 28483 df-wwlksn 28484 |
This theorem is referenced by: hashwwlksnext 28567 |
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