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Theorem numclwwlkovq 29026
Description: Value of operation 𝑄, mapping a vertex 𝑣 and a positive integer 𝑛 to the not closed walks v(0) ... v(n) of length 𝑛 from a fixed vertex 𝑣 = v(0). "Not closed" means v(n) =/= v(0). Remark: 𝑛 ∈ β„•0 would not be useful: numclwwlkqhash 29027 would not hold, because (𝐾↑0) = 1! (Contributed by Alexander van der Vekens, 27-Sep-2018.) (Revised by AV, 30-May-2021.)
Hypotheses
Ref Expression
numclwwlk.v 𝑉 = (Vtxβ€˜πΊ)
numclwwlk.q 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})
Assertion
Ref Expression
numclwwlkovq ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋𝑄𝑁) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)})
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑀
Allowed substitution hints:   𝑄(𝑀,𝑣,𝑛)   𝑉(𝑀)

Proof of Theorem numclwwlkovq
StepHypRef Expression
1 oveq1 7344 . . . 4 (𝑛 = 𝑁 β†’ (𝑛 WWalksN 𝐺) = (𝑁 WWalksN 𝐺))
21adantl 482 . . 3 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ (𝑛 WWalksN 𝐺) = (𝑁 WWalksN 𝐺))
3 eqeq2 2748 . . . . 5 (𝑣 = 𝑋 β†’ ((π‘€β€˜0) = 𝑣 ↔ (π‘€β€˜0) = 𝑋))
4 neeq2 3004 . . . . 5 (𝑣 = 𝑋 β†’ ((lastSβ€˜π‘€) β‰  𝑣 ↔ (lastSβ€˜π‘€) β‰  𝑋))
53, 4anbi12d 631 . . . 4 (𝑣 = 𝑋 β†’ (((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣) ↔ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)))
65adantr 481 . . 3 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ (((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣) ↔ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)))
72, 6rabeqbidv 3420 . 2 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)})
8 numclwwlk.q . 2 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})
9 ovex 7370 . . 3 (𝑁 WWalksN 𝐺) ∈ V
109rabex 5276 . 2 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)} ∈ V
117, 8, 10ovmpoa 7490 1 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋𝑄𝑁) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105   β‰  wne 2940  {crab 3403  β€˜cfv 6479  (class class class)co 7337   ∈ cmpo 7339  0cc0 10972  β„•cn 12074  lastSclsw 14365  Vtxcvtx 27655   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-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  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-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342
This theorem is referenced by:  numclwwlkqhash  29027  numclwwlk2lem1  29028  numclwlk2lem2f  29029  numclwlk2lem2f1o  29031
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