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Theorem wspthsnon 29106
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 29088 . . 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 6892 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
4 wwlksnon.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
53, 4eqtr4di 2791 . . . . 5 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = 𝑉)
65adantl 483 . . . 4 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ (Vtxβ€˜π‘”) = 𝑉)
7 oveq12 7418 . . . . . 6 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ (𝑛 WWalksNOn 𝑔) = (𝑁 WWalksNOn 𝐺))
87oveqd 7426 . . . . 5 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) = (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏))
9 fveq2 6892 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (SPathsOnβ€˜π‘”) = (SPathsOnβ€˜πΊ))
109oveqd 7426 . . . . . . . 8 (𝑔 = 𝐺 β†’ (π‘Ž(SPathsOnβ€˜π‘”)𝑏) = (π‘Ž(SPathsOnβ€˜πΊ)𝑏))
1110breqd 5160 . . . . . . 7 (𝑔 = 𝐺 β†’ (𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀 ↔ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀))
1211adantl 483 . . . . . 6 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ (𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀 ↔ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀))
1312exbidv 1925 . . . . 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 3450 . . . 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 7484 . . 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 483 . 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 484 . 2 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ π‘ˆ) β†’ 𝑁 ∈ β„•0)
18 elex 3493 . . 3 (𝐺 ∈ π‘ˆ β†’ 𝐺 ∈ V)
1918adantl 483 . 2 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ π‘ˆ) β†’ 𝐺 ∈ V)
204fvexi 6906 . . . 4 𝑉 ∈ V
2120, 20mpoex 8066 . . 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 7560 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 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {crab 3433  Vcvv 3475   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  β„•0cn0 12472  Vtxcvtx 28256  SPathsOncspthson 28972   WWalksNOn cwwlksnon 29081   WSPathsNOn cwwspthsnon 29083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-wspthsnon 29088
This theorem is referenced by:  iswspthsnon  29110
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