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Theorem wspthsn 41046
 Description: The set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
Assertion
Ref Expression
wspthsn (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤}
Distinct variable groups:   𝑓,𝐺,𝑤   𝑤,𝑁
Allowed substitution hint:   𝑁(𝑓)

Proof of Theorem wspthsn
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 6558 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑛 WWalkSN 𝑔) = (𝑁 WWalkSN 𝐺))
2 fveq2 6103 . . . . . . 7 (𝑔 = 𝐺 → (SPathS‘𝑔) = (SPathS‘𝐺))
32breqd 4594 . . . . . 6 (𝑔 = 𝐺 → (𝑓(SPathS‘𝑔)𝑤𝑓(SPathS‘𝐺)𝑤))
43exbidv 1837 . . . . 5 (𝑔 = 𝐺 → (∃𝑓 𝑓(SPathS‘𝑔)𝑤 ↔ ∃𝑓 𝑓(SPathS‘𝐺)𝑤))
54adantl 481 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → (∃𝑓 𝑓(SPathS‘𝑔)𝑤 ↔ ∃𝑓 𝑓(SPathS‘𝐺)𝑤))
61, 5rabeqbidv 3168 . . 3 ((𝑛 = 𝑁𝑔 = 𝐺) → {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ∃𝑓 𝑓(SPathS‘𝑔)𝑤} = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤})
7 df-wspthsn 41036 . . 3 WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ∃𝑓 𝑓(SPathS‘𝑔)𝑤})
8 ovex 6577 . . . 4 (𝑁 WWalkSN 𝐺) ∈ V
98rabex 4740 . . 3 {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤} ∈ V
106, 7, 9ovmpt2a 6689 . 2 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤})
117mpt2ndm0 6773 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = ∅)
12 df-wwlksn 41034 . . . . . 6 WWalkSN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalkS‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)})
1312mpt2ndm0 6773 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalkSN 𝐺) = ∅)
1413rabeqdv 3167 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤} = {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤})
15 rab0 3909 . . . 4 {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤} = ∅
1614, 15syl6eq 2660 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤} = ∅)
1711, 16eqtr4d 2647 . 2 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤})
1810, 17pm2.61i 175 1 (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900  Vcvv 3173  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818  ℕ0cn0 11169  #chash 12979  SPathScspths 40920  WWalkScwwlks 41028   WWalkSN cwwlksn 41029   WSPathsN cwwspthsn 41031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wwlksn 41034  df-wspthsn 41036 This theorem is referenced by:  iswspthn  41047  wspn0  41131
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