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Theorem wwlksnexthasheq 28556
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.)
Hypotheses
Ref Expression
wwlksnexthasheq.v 𝑉 = (Vtxβ€˜πΊ)
wwlksnexthasheq.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
wwlksnexthasheq (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}) = (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸}))
Distinct variable groups:   𝑛,𝐸,𝑀   𝑀,𝐺   𝑀,𝑁   𝑛,𝑉,𝑀   𝑛,π‘Š,𝑀
Allowed substitution hints:   𝐺(𝑛)   𝑁(𝑛)

Proof of Theorem wwlksnexthasheq
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ovex 7370 . . 3 ((𝑁 + 1) WWalksN 𝐺) ∈ V
21rabex 5276 . 2 {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)} ∈ V
3 wwlksnexthasheq.v . . 3 𝑉 = (Vtxβ€˜πΊ)
4 wwlksnexthasheq.e . . 3 𝐸 = (Edgβ€˜πΊ)
53, 4wwlksnextbij 28555 . 2 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ βˆƒπ‘“ 𝑓:{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}–1-1-ontoβ†’{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸})
6 hasheqf1oi 14166 . 2 ({𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)} ∈ V β†’ (βˆƒπ‘“ 𝑓:{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}–1-1-ontoβ†’{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸} β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}) = (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸})))
72, 5, 6mpsyl 68 1 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}) = (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1540  βˆƒwex 1780   ∈ wcel 2105  {crab 3403  Vcvv 3441  {cpr 4575  β€“1-1-ontoβ†’wf1o 6478  β€˜cfv 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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