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Theorem wwlksn 29771
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 6901 . . . . . 6 (𝑔 = 𝐺 → (WWalks‘𝑔) = (WWalks‘𝐺))
21adantl 480 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → (WWalks‘𝑔) = (WWalks‘𝐺))
3 oveq1 7431 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
43eqeq2d 2737 . . . . . 6 (𝑛 = 𝑁 → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1)))
54adantr 479 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1)))
62, 5rabeqbidv 3437 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
7 df-wwlksn 29765 . . . 4 WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
8 fvex 6914 . . . . 5 (WWalks‘𝐺) ∈ V
98rabex 5339 . . . 4 {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} ∈ V
106, 7, 9ovmpoa 7581 . . 3 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
1110expcom 412 . 2 (𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
127reldmmpo 7560 . . . . 5 Rel dom WWalksN
1312ovprc2 7464 . . . 4 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = ∅)
14 fvprc 6893 . . . . . 6 𝐺 ∈ V → (WWalks‘𝐺) = ∅)
1514rabeqdv 3435 . . . . 5 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)})
16 rab0 4387 . . . . 5 {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅
1715, 16eqtrdi 2782 . . . 4 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅)
1813, 17eqtr4d 2769 . . 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 205  wa 394   = wceq 1534  wcel 2099  {crab 3419  Vcvv 3462  c0 4325  cfv 6554  (class class class)co 7424  1c1 11159   + caddc 11161  0cn0 12524  chash 14347  WWalkscwwlks 29759   WWalksN cwwlksn 29760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-wwlksn 29765
This theorem is referenced by:  iswwlksn  29772  wwlksn0s  29795  0enwwlksnge1  29798  wlknwwlksnbij  29822  wwlksnfi  29840
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