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Theorem wwlksn 27723
 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 6659 . . . . . 6 (𝑔 = 𝐺 → (WWalks‘𝑔) = (WWalks‘𝐺))
21adantl 486 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → (WWalks‘𝑔) = (WWalks‘𝐺))
3 oveq1 7158 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
43eqeq2d 2770 . . . . . 6 (𝑛 = 𝑁 → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1)))
54adantr 485 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1)))
62, 5rabeqbidv 3399 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
7 df-wwlksn 27717 . . . 4 WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
8 fvex 6672 . . . . 5 (WWalks‘𝐺) ∈ V
98rabex 5203 . . . 4 {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} ∈ V
106, 7, 9ovmpoa 7301 . . 3 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
1110expcom 418 . 2 (𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
127reldmmpo 7281 . . . . 5 Rel dom WWalksN
1312ovprc2 7191 . . . 4 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = ∅)
14 fvprc 6651 . . . . . 6 𝐺 ∈ V → (WWalks‘𝐺) = ∅)
1514rabeqdv 3398 . . . . 5 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)})
16 rab0 4280 . . . . 5 {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅
1715, 16eqtrdi 2810 . . . 4 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅)
1813, 17eqtr4d 2797 . . 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 185 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 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {crab 3075  Vcvv 3410  ∅c0 4226  ‘cfv 6336  (class class class)co 7151  1c1 10577   + caddc 10579  ℕ0cn0 11935  ♯chash 13741  WWalkscwwlks 27711   WWalksN cwwlksn 27712 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-wwlksn 27717 This theorem is referenced by:  iswwlksn  27724  wwlksn0s  27747  0enwwlksnge1  27750  wlknwwlksnbij  27774  wwlksnfi  27792
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