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Theorem clwwlkn 26295
 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.)
Assertion
Ref Expression
clwwlkn ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁})
Distinct variable groups:   𝑤,𝐸   𝑤,𝑉   𝑤,𝑁
Allowed substitution hints:   𝑋(𝑤)   𝑌(𝑤)

Proof of Theorem clwwlkn
Dummy variables 𝑒 𝑛 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1056 . . 3 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
2 ovex 6577 . . . 4 (𝑉 ClWWalks 𝐸) ∈ V
3 rabexg 4739 . . . 4 ((𝑉 ClWWalks 𝐸) ∈ V → {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁} ∈ V)
42, 3mp1i 13 . . 3 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁} ∈ V)
5 eqeq2 2621 . . . . 5 (𝑛 = 𝑁 → ((#‘𝑤) = 𝑛 ↔ (#‘𝑤) = 𝑁))
65rabbidv 3164 . . . 4 (𝑛 = 𝑁 → {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁})
7 eqid 2610 . . . 4 (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})
86, 7fvmptg 6189 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁} ∈ V) → ((𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁})
91, 4, 8syl2anc 691 . 2 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁})
10 df-clwwlkn 26280 . . . . . . 7 ClWWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}))
1110a1i 11 . . . . . 6 ((𝑉𝑋𝐸𝑌) → ClWWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛})))
12 oveq12 6558 . . . . . . . . 9 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 ClWWalks 𝑒) = (𝑉 ClWWalks 𝐸))
13 rabeq 3166 . . . . . . . . 9 ((𝑣 ClWWalks 𝑒) = (𝑉 ClWWalks 𝐸) → {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})
1412, 13syl 17 . . . . . . . 8 ((𝑣 = 𝑉𝑒 = 𝐸) → {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})
1514mpteq2dv 4673 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}))
1615adantl 481 . . . . . 6 (((𝑉𝑋𝐸𝑌) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}))
17 elex 3185 . . . . . . 7 (𝑉𝑋𝑉 ∈ V)
1817adantr 480 . . . . . 6 ((𝑉𝑋𝐸𝑌) → 𝑉 ∈ V)
19 elex 3185 . . . . . . 7 (𝐸𝑌𝐸 ∈ V)
2019adantl 481 . . . . . 6 ((𝑉𝑋𝐸𝑌) → 𝐸 ∈ V)
21 nn0ex 11175 . . . . . . . 8 0 ∈ V
2221mptex 6390 . . . . . . 7 (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}) ∈ V
2322a1i 11 . . . . . 6 ((𝑉𝑋𝐸𝑌) → (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}) ∈ V)
2411, 16, 18, 20, 23ovmpt2d 6686 . . . . 5 ((𝑉𝑋𝐸𝑌) → (𝑉 ClWWalksN 𝐸) = (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}))
2524fveq1d 6105 . . . 4 ((𝑉𝑋𝐸𝑌) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = ((𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})‘𝑁))
2625eqeq1d 2612 . . 3 ((𝑉𝑋𝐸𝑌) → (((𝑉 ClWWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁} ↔ ((𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁}))
27263adant3 1074 . 2 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → (((𝑉 ClWWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁} ↔ ((𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁}))
289, 27mpbird 246 1 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ℕ0cn0 11169  #chash 12979   ClWWalks cclwwlk 26276   ClWWalksN cclwwlkn 26277 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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898  df-n0 11170  df-clwwlkn 26280 This theorem is referenced by:  isclwwlkn  26297  clwwlknprop  26300  clwwlkn0  26302  clwwlknfi  26306
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