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Theorem wspthsn 29868
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 7440 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑛 WWalksN 𝑔) = (𝑁 WWalksN 𝐺))
2 fveq2 6906 . . . . . . 7 (𝑔 = 𝐺 → (SPaths‘𝑔) = (SPaths‘𝐺))
32breqd 5154 . . . . . 6 (𝑔 = 𝐺 → (𝑓(SPaths‘𝑔)𝑤𝑓(SPaths‘𝐺)𝑤))
43exbidv 1921 . . . . 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 3455 . . 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 29853 . . 3 WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
8 ovex 7464 . . . 4 (𝑁 WWalksN 𝐺) ∈ V
98rabex 5339 . . 3 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} ∈ V
106, 7, 9ovmpoa 7588 . 2 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
117mpondm0 7673 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = ∅)
12 df-wwlksn 29851 . . . . . 6 WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
1312mpondm0 7673 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
1413rabeqdv 3452 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
15 rab0 4386 . . . 4 {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅
1614, 15eqtrdi 2793 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅)
1711, 16eqtr4d 2780 . 2 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
1810, 17pm2.61i 182 1 (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  {crab 3436  Vcvv 3480  c0 4333   class class class wbr 5143  cfv 6561  (class class class)co 7431  1c1 11156   + caddc 11158  0cn0 12526  chash 14369  SPathscspths 29731  WWalkscwwlks 29845   WWalksN cwwlksn 29846   WSPathsN cwwspthsn 29848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-wwlksn 29851  df-wspthsn 29853
This theorem is referenced by:  iswspthn  29869  wspn0  29944
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