Theorem iswwlksn 30187

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 30186	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
2	1	eleq2d 2849	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ 𝑊 ∈ {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
3		fveqeq2 6890	. . 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 290	1 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416  ‘cfv 6536  (class class class)co 7410  1c1 11096   + caddc 11098  ℕ₀cn0 12499  ♯chash 14362  WWalkscwwlks 30174   WWalksN cwwlksn 30175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-wwlksn 30180

This theorem is referenced by:  wwlksnprcl  30188  iswwlksnx  30189  wwlknbp  30191  wwlknp  30192  wwlkswwlksn  30214  wlklnwwlkln1  30217  wlklnwwlkln2lem  30231  wlknewwlksn  30236  wwlksnred  30241  wwlksnext  30242  wwlksnextproplem3  30260  wspthsnonn0vne  30266  elwspths2spth  30319  rusgrnumwwlkl1  30320  clwwlkel  30397  clwwlkf  30398  clwwlknwwlksnb  30406