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Theorem numclwwlkovq 30223
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 30224 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 7420 . . . 4 (𝑛 = 𝑁 β†’ (𝑛 WWalksN 𝐺) = (𝑁 WWalksN 𝐺))
21adantl 480 . . 3 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ (𝑛 WWalksN 𝐺) = (𝑁 WWalksN 𝐺))
3 eqeq2 2737 . . . . 5 (𝑣 = 𝑋 β†’ ((π‘€β€˜0) = 𝑣 ↔ (π‘€β€˜0) = 𝑋))
4 neeq2 2994 . . . . 5 (𝑣 = 𝑋 β†’ ((lastSβ€˜π‘€) β‰  𝑣 ↔ (lastSβ€˜π‘€) β‰  𝑋))
53, 4anbi12d 630 . . . 4 (𝑣 = 𝑋 β†’ (((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣) ↔ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)))
65adantr 479 . . 3 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ (((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣) ↔ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)))
72, 6rabeqbidv 3437 . 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 7446 . . 3 (𝑁 WWalksN 𝐺) ∈ V
109rabex 5330 . 2 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)} ∈ V
117, 8, 10ovmpoa 7570 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 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  {crab 3419  β€˜cfv 6543  (class class class)co 7413   ∈ cmpo 7415  0cc0 11133  β„•cn 12237  lastSclsw 14539  Vtxcvtx 28848   WWalksN cwwlksn 29676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418
This theorem is referenced by:  numclwwlkqhash  30224  numclwwlk2lem1  30225  numclwlk2lem2f  30226  numclwlk2lem2f1o  30228
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