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Theorem wwlksn 29867
Description: The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
Assertion
Ref Expression
wwlksn (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁

Proof of Theorem wwlksn
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . . . 6 (𝑔 = 𝐺 → (WWalks‘𝑔) = (WWalks‘𝐺))
21adantl 481 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → (WWalks‘𝑔) = (WWalks‘𝐺))
3 oveq1 7438 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
43eqeq2d 2746 . . . . . 6 (𝑛 = 𝑁 → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1)))
54adantr 480 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1)))
62, 5rabeqbidv 3452 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
7 df-wwlksn 29861 . . . 4 WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
8 fvex 6920 . . . . 5 (WWalks‘𝐺) ∈ V
98rabex 5345 . . . 4 {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} ∈ V
106, 7, 9ovmpoa 7588 . . 3 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
1110expcom 413 . 2 (𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
127reldmmpo 7567 . . . . 5 Rel dom WWalksN
1312ovprc2 7471 . . . 4 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = ∅)
14 fvprc 6899 . . . . . 6 𝐺 ∈ V → (WWalks‘𝐺) = ∅)
1514rabeqdv 3449 . . . . 5 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)})
16 rab0 4392 . . . . 5 {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅
1715, 16eqtrdi 2791 . . . 4 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅)
1813, 17eqtr4d 2778 . . 3 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
1918a1d 25 . 2 𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
2011, 19pm2.61i 182 1 (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  c0 4339  cfv 6563  (class class class)co 7431  1c1 11154   + caddc 11156  0cn0 12524  chash 14366  WWalkscwwlks 29855   WWalksN cwwlksn 29856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-wwlksn 29861
This theorem is referenced by:  iswwlksn  29868  wwlksn0s  29891  0enwwlksnge1  29894  wlknwwlksnbij  29918  wwlksnfi  29936
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