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Theorem isclwwlksn 26876
 Description: A word over the set of vertices representing a closed walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.)
Assertion
Ref Expression
isclwwlksn (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (#‘𝑊) = 𝑁)))

Proof of Theorem isclwwlksn
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 clwwlksn 26875 . . 3 (𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})
21eleq2d 2686 . 2 (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ 𝑊 ∈ {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁}))
3 fveq2 6189 . . . 4 (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
43eqeq1d 2623 . . 3 (𝑤 = 𝑊 → ((#‘𝑤) = 𝑁 ↔ (#‘𝑊) = 𝑁))
54elrab 3361 . 2 (𝑊 ∈ {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁} ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (#‘𝑊) = 𝑁))
62, 5syl6bb 276 1 (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (#‘𝑊) = 𝑁)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1482   ∈ wcel 1989  {crab 2915  ‘cfv 5886  (class class class)co 6647  ℕcn 11017  #chash 13112  ClWWalkscclwwlks 26869   ClWWalksN cclwwlksn 26870 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-clwwlksn 26872 This theorem is referenced by:  clwwlknbp0  26878  clwwlknp  26881  isclwwlksng  26882  clwwlkclwwlkn  26885  clwwnisshclwwsn  26923  clwlksfclwwlk  26955  clwlksfoclwwlk  26956  extwwlkfablem2  27197
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