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Theorem wspthsnon 29144
Description: The set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
Hypothesis
Ref Expression
wwlksnon.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
wspthsnon ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ π‘ˆ) β†’ (𝑁 WSPathsNOn 𝐺) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀}))
Distinct variable groups:   𝐺,π‘Ž,𝑏,𝑀   𝑁,π‘Ž,𝑏,𝑀   𝑉,π‘Ž,𝑏   𝑓,𝐺,π‘Ž,𝑏,𝑀   𝑓,𝑁
Allowed substitution hints:   π‘ˆ(𝑀,𝑓,π‘Ž,𝑏)   𝑉(𝑀,𝑓)
Proof of Theorem wspthsnon
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wspthsnon 29126 . . 3 WSPathsNOn = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀}))
21a1i 11 . 2 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ π‘ˆ) β†’ WSPathsNOn = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀})))
3 fveq2 6891 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
4 wwlksnon.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
53, 4eqtr4di 2790 . . . . 5 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = 𝑉)
65adantl 482 . . . 4 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ (Vtxβ€˜π‘”) = 𝑉)
7 oveq12 7420 . . . . . 6 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ (𝑛 WWalksNOn 𝑔) = (𝑁 WWalksNOn 𝐺))
87oveqd 7428 . . . . 5 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) = (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏))
9 fveq2 6891 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (SPathsOnβ€˜π‘”) = (SPathsOnβ€˜πΊ))
109oveqd 7428 . . . . . . . 8 (𝑔 = 𝐺 β†’ (π‘Ž(SPathsOnβ€˜π‘”)𝑏) = (π‘Ž(SPathsOnβ€˜πΊ)𝑏))
1110breqd 5159 . . . . . . 7 (𝑔 = 𝐺 β†’ (𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀 ↔ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀))
1211adantl 482 . . . . . 6 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ (𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀 ↔ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀))
1312exbidv 1924 . . . . 5 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ (βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀 ↔ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀))
148, 13rabeqbidv 3449 . . . 4 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀} = {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀})
156, 6, 14mpoeq123dv 7486 . . 3 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀}) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀}))
1615adantl 482 . 2 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ π‘ˆ) ∧ (𝑛 = 𝑁 ∧ 𝑔 = 𝐺)) β†’ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀}) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀}))
17 simpl 483 . 2 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ π‘ˆ) β†’ 𝑁 ∈ β„•0)
18 elex 3492 . . 3 (𝐺 ∈ π‘ˆ β†’ 𝐺 ∈ V)
1918adantl 482 . 2 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ π‘ˆ) β†’ 𝐺 ∈ V)
204fvexi 6905 . . . 4 𝑉 ∈ V
2120, 20mpoex 8068 . . 3 (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀}) ∈ V
2221a1i 11 . 2 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ π‘ˆ) β†’ (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀}) ∈ V)
232, 16, 17, 19, 22ovmpod 7562 1 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ π‘ˆ) β†’ (𝑁 WSPathsNOn 𝐺) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {crab 3432  Vcvv 3474   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413  β„•0cn0 12474  Vtxcvtx 28294  SPathsOncspthson 29010   WWalksNOn cwwlksnon 29119   WSPathsNOn cwwspthsnon 29121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-wspthsnon 29126
This theorem is referenced by:  iswspthsnon  29148
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