Theorem iswwlksn 30038

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.

Step	Hyp	Ref	Expression
1		wwlksn 30037	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
2	1	eleq2d 2848	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ 𝑊 ∈ {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
3		fveqeq2 6876	. . 3 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) = (𝑁 + 1) ↔ (♯‘𝑊) = (𝑁 + 1)))
4	3	elrab 3650	. 2 ⊢ (𝑊 ∈ {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))
5	2, 4	bitrdi 289	1 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {crab 3414  ‘cfv 6521  (class class class)co 7396  1c1 11074   + caddc 11076  ℕ₀cn0 12481  ♯chash 14343  WWalkscwwlks 30025   WWalksN cwwlksn 30026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-wwlksn 30031

This theorem is referenced by:  wwlksnprcl  30039  iswwlksnx  30040  wwlknbp  30042  wwlknp  30043  wwlkswwlksn  30065  wlklnwwlkln1  30068  wlklnwwlkln2lem  30082  wlknewwlksn  30087  wwlksnred  30092  wwlksnext  30093  wwlksnextproplem3  30111  wspthsnonn0vne  30117  elwspths2spth  30170  rusgrnumwwlkl1  30171  clwwlkel  30248  clwwlkf  30249  clwwlknwwlksnb  30257