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Theorem wspthsnon 27648
 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 = (Vtx‘𝐺)
Assertion
Ref Expression
wspthsnon ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
Distinct variable groups:   𝐺,𝑎,𝑏,𝑤   𝑁,𝑎,𝑏,𝑤   𝑉,𝑎,𝑏   𝑓,𝐺,𝑎,𝑏,𝑤   𝑓,𝑁
Allowed substitution hints:   𝑈(𝑤,𝑓,𝑎,𝑏)   𝑉(𝑤,𝑓)

Proof of Theorem wspthsnon
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wspthsnon 27630 . . 3 WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))
21a1i 11 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤})))
3 fveq2 6646 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4 wwlksnon.v . . . . . 6 𝑉 = (Vtx‘𝐺)
53, 4eqtr4di 2851 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
65adantl 485 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → (Vtx‘𝑔) = 𝑉)
7 oveq12 7145 . . . . . 6 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑛 WWalksNOn 𝑔) = (𝑁 WWalksNOn 𝐺))
87oveqd 7153 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑎(𝑛 WWalksNOn 𝑔)𝑏) = (𝑎(𝑁 WWalksNOn 𝐺)𝑏))
9 fveq2 6646 . . . . . . . . 9 (𝑔 = 𝐺 → (SPathsOn‘𝑔) = (SPathsOn‘𝐺))
109oveqd 7153 . . . . . . . 8 (𝑔 = 𝐺 → (𝑎(SPathsOn‘𝑔)𝑏) = (𝑎(SPathsOn‘𝐺)𝑏))
1110breqd 5042 . . . . . . 7 (𝑔 = 𝐺 → (𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤))
1211adantl 485 . . . . . 6 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤))
1312exbidv 1922 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → (∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤 ↔ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤))
148, 13rabeqbidv 3433 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤} = {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})
156, 6, 14mpoeq123dv 7209 . . 3 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
1615adantl 485 . 2 (((𝑁 ∈ ℕ0𝐺𝑈) ∧ (𝑛 = 𝑁𝑔 = 𝐺)) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
17 simpl 486 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → 𝑁 ∈ ℕ0)
18 elex 3459 . . 3 (𝐺𝑈𝐺 ∈ V)
1918adantl 485 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → 𝐺 ∈ V)
204fvexi 6660 . . . 4 𝑉 ∈ V
2120, 20mpoex 7763 . . 3 (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}) ∈ V
2221a1i 11 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}) ∈ V)
232, 16, 17, 19, 22ovmpod 7283 1 ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {crab 3110  Vcvv 3441   class class class wbr 5031  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  ℕ0cn0 11888  Vtxcvtx 26799  SPathsOncspthson 27514   WWalksNOn cwwlksnon 27623   WSPathsNOn cwwspthsnon 27625 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-wspthsnon 27630 This theorem is referenced by:  iswspthsnon  27652
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