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Theorem clwwlksn 41189
 Description: The set of closed walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)
Assertion
Ref Expression
clwwlksn (𝑁 ∈ ℕ → (𝑁 ClWWalkSN 𝐺) = {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁})
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁

Proof of Theorem clwwlksn
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-clwwlksn 41186 . . . . 5 ClWWalkSN = (𝑛 ∈ ℕ, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalkS‘𝑔) ∣ (#‘𝑤) = 𝑛})
21a1i 11 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → ClWWalkSN = (𝑛 ∈ ℕ, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalkS‘𝑔) ∣ (#‘𝑤) = 𝑛}))
3 fveq2 6103 . . . . . . 7 (𝑔 = 𝐺 → (ClWWalkS‘𝑔) = (ClWWalkS‘𝐺))
43adantl 481 . . . . . 6 ((𝑛 = 𝑁𝑔 = 𝐺) → (ClWWalkS‘𝑔) = (ClWWalkS‘𝐺))
5 eqeq2 2621 . . . . . . 7 (𝑛 = 𝑁 → ((#‘𝑤) = 𝑛 ↔ (#‘𝑤) = 𝑁))
65adantr 480 . . . . . 6 ((𝑛 = 𝑁𝑔 = 𝐺) → ((#‘𝑤) = 𝑛 ↔ (#‘𝑤) = 𝑁))
74, 6rabeqbidv 3168 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → {𝑤 ∈ (ClWWalkS‘𝑔) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁})
87adantl 481 . . . 4 (((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) ∧ (𝑛 = 𝑁𝑔 = 𝐺)) → {𝑤 ∈ (ClWWalkS‘𝑔) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁})
9 simpl 472 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → 𝑁 ∈ ℕ)
10 simpr 476 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → 𝐺 ∈ V)
11 fvex 6113 . . . . . 6 (ClWWalkS‘𝐺) ∈ V
1211rabex 4740 . . . . 5 {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁} ∈ V
1312a1i 11 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁} ∈ V)
142, 8, 9, 10, 13ovmpt2d 6686 . . 3 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → (𝑁 ClWWalkSN 𝐺) = {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁})
1514expcom 450 . 2 (𝐺 ∈ V → (𝑁 ∈ ℕ → (𝑁 ClWWalkSN 𝐺) = {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁}))
161reldmmpt2 6669 . . . . 5 Rel dom ClWWalkSN
1716ovprc2 6583 . . . 4 𝐺 ∈ V → (𝑁 ClWWalkSN 𝐺) = ∅)
18 fvprc 6097 . . . . . 6 𝐺 ∈ V → (ClWWalkS‘𝐺) = ∅)
1918rabeqdv 3167 . . . . 5 𝐺 ∈ V → {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁} = {𝑤 ∈ ∅ ∣ (#‘𝑤) = 𝑁})
20 rab0 3909 . . . . 5 {𝑤 ∈ ∅ ∣ (#‘𝑤) = 𝑁} = ∅
2119, 20syl6eq 2660 . . . 4 𝐺 ∈ V → {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁} = ∅)
2217, 21eqtr4d 2647 . . 3 𝐺 ∈ V → (𝑁 ClWWalkSN 𝐺) = {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁})
2322a1d 25 . 2 𝐺 ∈ V → (𝑁 ∈ ℕ → (𝑁 ClWWalkSN 𝐺) = {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁}))
2415, 23pm2.61i 175 1 (𝑁 ∈ ℕ → (𝑁 ClWWalkSN 𝐺) = {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ∅c0 3874  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ℕcn 10897  #chash 12979  ClWWalkScclwwlks 41183   ClWWalkSN cclwwlksn 41184 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-clwwlksn 41186 This theorem is referenced by:  isclwwlksn  41190  clwwlksnfi  41220
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