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Theorem clwwlksn 26782
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 26779 . . . . 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 6158 . . . . . . 7 (𝑔 = 𝐺 → (ClWWalks‘𝑔) = (ClWWalks‘𝐺))
43adantl 482 . . . . . 6 ((𝑛 = 𝑁𝑔 = 𝐺) → (ClWWalks‘𝑔) = (ClWWalks‘𝐺))
5 eqeq2 2632 . . . . . . 7 (𝑛 = 𝑁 → ((#‘𝑤) = 𝑛 ↔ (#‘𝑤) = 𝑁))
65adantr 481 . . . . . 6 ((𝑛 = 𝑁𝑔 = 𝐺) → ((#‘𝑤) = 𝑛 ↔ (#‘𝑤) = 𝑁))
74, 6rabeqbidv 3185 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → {𝑤 ∈ (ClWWalks‘𝑔) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})
87adantl 482 . . . 4 (((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) ∧ (𝑛 = 𝑁𝑔 = 𝐺)) → {𝑤 ∈ (ClWWalks‘𝑔) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})
9 simpl 473 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → 𝑁 ∈ ℕ)
10 simpr 477 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → 𝐺 ∈ V)
11 fvex 6168 . . . . . 6 (ClWWalks‘𝐺) ∈ V
1211rabex 4783 . . . . 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 6753 . . 3 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})
1514expcom 451 . 2 (𝐺 ∈ V → (𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁}))
161reldmmpt2 6736 . . . . 5 Rel dom ClWWalksN
1716ovprc2 6650 . . . 4 𝐺 ∈ V → (𝑁 ClWWalksN 𝐺) = ∅)
18 fvprc 6152 . . . . . 6 𝐺 ∈ V → (ClWWalks‘𝐺) = ∅)
1918rabeqdv 3184 . . . . 5 𝐺 ∈ V → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁} = {𝑤 ∈ ∅ ∣ (#‘𝑤) = 𝑁})
20 rab0 3935 . . . . 5 {𝑤 ∈ ∅ ∣ (#‘𝑤) = 𝑁} = ∅
2119, 20syl6eq 2671 . . . 4 𝐺 ∈ V → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁} = ∅)
2217, 21eqtr4d 2658 . . 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 176 1 (𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {crab 2912  Vcvv 3190  c0 3897  cfv 5857  (class class class)co 6615  cmpt2 6617  cn 10980  #chash 13073  ClWWalkscclwwlks 26776   ClWWalksN cclwwlksn 26777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-clwwlksn 26779
This theorem is referenced by:  isclwwlksn  26783  clwwlksnfi  26813
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