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Theorem iswwlksn 26786
Description: A word over the set of vertices representing a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
Assertion
Ref Expression
iswwlksn (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))))

Proof of Theorem iswwlksn
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 wwlksn 26785 . . 3 (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})
21eleq2d 2716 . 2 (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ 𝑊 ∈ {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)}))
3 fveq2 6229 . . . 4 (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
43eqeq1d 2653 . . 3 (𝑤 = 𝑊 → ((#‘𝑤) = (𝑁 + 1) ↔ (#‘𝑊) = (𝑁 + 1)))
54elrab 3396 . 2 (𝑊 ∈ {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)} ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)))
62, 5syl6bb 276 1 (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {crab 2945  cfv 5926  (class class class)co 6690  1c1 9975   + caddc 9977  0cn0 11330  #chash 13157  WWalkscwwlks 26773   WWalksN cwwlksn 26774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-wwlksn 26779
This theorem is referenced by:  wwlksnprcl  26787  iswwlksnx  26788  wwlknbp  26790  wwlknp  26791  wwlkswwlksn  26819  wlklnwwlkln1  26822  wlklnwwlkln2lem  26836  wlknewwlksn  26841  wwlksnred  26855  wwlksnext  26856  wwlksnextproplem3  26874  wspthsnonn0vne  26882  elwspths2spth  26934  rusgrnumwwlkl1  26935  clwwlkel  27009  clwwlkf  27010  clwwlknwwlksnb  27019
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