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Theorem iswwlksn 26862
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 26861 . . 3 (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
21eleq2d 2789 . 2 (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ 𝑊 ∈ {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
3 fveq2 6304 . . . 4 (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
43eqeq1d 2726 . . 3 (𝑤 = 𝑊 → ((♯‘𝑤) = (𝑁 + 1) ↔ (♯‘𝑊) = (𝑁 + 1)))
54elrab 3469 . 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 1596  wcel 2103  {crab 3018  cfv 6001  (class class class)co 6765  1c1 10050   + caddc 10052  0cn0 11405  chash 13232  WWalkscwwlks 26849   WWalksN cwwlksn 26850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-wwlksn 26855
This theorem is referenced by:  wwlksnprcl  26863  iswwlksnx  26864  wwlknbp  26866  wwlknp  26867  wwlkswwlksn  26895  wlklnwwlkln1  26898  wlklnwwlkln2lem  26912  wlknewwlksn  26917  wwlksnred  26931  wwlksnext  26932  wwlksnextproplem3  26950  wspthsnonn0vne  26958  elwspths2spth  27010  rusgrnumwwlkl1  27011  clwwlkel  27096  clwwlkf  27097  clwwlknwwlksnb  27106
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