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Mirrors > Home > MPE Home > Th. List > wwlksnexthasheq | Structured version Visualization version GIF version |
Description: The number of the extensions of a walk (as word) by an edge equals the number of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 23-Aug-2018.) (Revised by AV, 19-Apr-2021.) (Revised by AV, 27-Oct-2022.) |
Ref | Expression |
---|---|
wwlksnexthasheq.v | β’ π = (VtxβπΊ) |
wwlksnexthasheq.e | β’ πΈ = (EdgβπΊ) |
Ref | Expression |
---|---|
wwlksnexthasheq | β’ (π β (π WWalksN πΊ) β (β―β{π€ β ((π + 1) WWalksN πΊ) β£ ((π€ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ€)} β πΈ)}) = (β―β{π β π β£ {(lastSβπ), π} β πΈ})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7370 | . . 3 β’ ((π + 1) WWalksN πΊ) β V | |
2 | 1 | rabex 5276 | . 2 β’ {π€ β ((π + 1) WWalksN πΊ) β£ ((π€ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ€)} β πΈ)} β V |
3 | wwlksnexthasheq.v | . . 3 β’ π = (VtxβπΊ) | |
4 | wwlksnexthasheq.e | . . 3 β’ πΈ = (EdgβπΊ) | |
5 | 3, 4 | wwlksnextbij 28555 | . 2 β’ (π β (π WWalksN πΊ) β βπ π:{π€ β ((π + 1) WWalksN πΊ) β£ ((π€ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ€)} β πΈ)}β1-1-ontoβ{π β π β£ {(lastSβπ), π} β πΈ}) |
6 | hasheqf1oi 14166 | . 2 β’ ({π€ β ((π + 1) WWalksN πΊ) β£ ((π€ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ€)} β πΈ)} β V β (βπ π:{π€ β ((π + 1) WWalksN πΊ) β£ ((π€ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ€)} β πΈ)}β1-1-ontoβ{π β π β£ {(lastSβπ), π} β πΈ} β (β―β{π€ β ((π + 1) WWalksN πΊ) β£ ((π€ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ€)} β πΈ)}) = (β―β{π β π β£ {(lastSβπ), π} β πΈ}))) | |
7 | 2, 5, 6 | mpsyl 68 | 1 β’ (π β (π WWalksN πΊ) β (β―β{π€ β ((π + 1) WWalksN πΊ) β£ ((π€ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ€)} β πΈ)}) = (β―β{π β π β£ {(lastSβπ), π} β πΈ})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1540 βwex 1780 β wcel 2105 {crab 3403 Vcvv 3441 {cpr 4575 β1-1-ontoβwf1o 6478 βcfv 6479 (class class class)co 7337 1c1 10973 + caddc 10975 β―chash 14145 lastSclsw 14365 prefix cpfx 14481 Vtxcvtx 27655 Edgcedg 27706 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-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-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-2 12137 df-n0 12335 df-xnn0 12407 df-z 12421 df-uz 12684 df-rp 12832 df-fz 13341 df-fzo 13484 df-hash 14146 df-word 14318 df-lsw 14366 df-concat 14374 df-s1 14400 df-substr 14452 df-pfx 14482 df-wwlks 28483 df-wwlksn 28484 |
This theorem is referenced by: rusgrnumwwlks 28627 |
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