Theorem iswwlksn 29768

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 29767	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
2	1	eleq2d 2814	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ 𝑊 ∈ {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
3		fveqeq2 6867	. . 3 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) = (𝑁 + 1) ↔ (♯‘𝑊) = (𝑁 + 1)))
4	3	elrab 3659	. 2 ⊢ (𝑊 ∈ {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))
5	2, 4	bitrdi 287	1 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3405  ‘cfv 6511  (class class class)co 7387  1c1 11069   + caddc 11071  ℕ₀cn0 12442  ♯chash 14295  WWalkscwwlks 29755   WWalksN cwwlksn 29756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-wwlksn 29761

This theorem is referenced by:  wwlksnprcl  29769  iswwlksnx  29770  wwlknbp  29772  wwlknp  29773  wwlkswwlksn  29795  wlklnwwlkln1  29798  wlklnwwlkln2lem  29812  wlknewwlksn  29817  wwlksnred  29822  wwlksnext  29823  wwlksnextproplem3  29841  wspthsnonn0vne  29847  elwspths2spth  29897  rusgrnumwwlkl1  29898  clwwlkel  29975  clwwlkf  29976  clwwlknwwlksnb  29984