Theorem isclwwlksn 41190

Description: A word over the set of vertices representing a closed walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.)

Assertion

Ref	Expression
isclwwlksn	⊢ (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalkSN 𝐺) ↔ (𝑊 ∈ (ClWWalkS‘𝐺) ∧ (#‘𝑊) = 𝑁)))

Proof of Theorem isclwwlksn

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwwlksn 41189	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 ClWWalkSN 𝐺) = {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁})
2	1	eleq2d 2673	. 2 ⊢ (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalkSN 𝐺) ↔ 𝑊 ∈ {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁}))
3		fveq2 6103	. . . 4 ⊢ (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
4	3	eqeq1d 2612	. . 3 ⊢ (𝑤 = 𝑊 → ((#‘𝑤) = 𝑁 ↔ (#‘𝑊) = 𝑁))
5	4	elrab 3331	. 2 ⊢ (𝑊 ∈ {𝑤 ∈ (ClWWalkS‘𝐺) ∣ (#‘𝑤) = 𝑁} ↔ (𝑊 ∈ (ClWWalkS‘𝐺) ∧ (#‘𝑊) = 𝑁))
6	2, 5	syl6bb 275	1 ⊢ (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalkSN 𝐺) ↔ (𝑊 ∈ (ClWWalkS‘𝐺) ∧ (#‘𝑊) = 𝑁)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  ‘cfv 5804  (class class class)co 6549  ℕcn 10897  #chash 12979  ClWWalkScclwwlks 41183   ClWWalkSN cclwwlksn 41184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-clwwlksn 41186

This theorem is referenced by:  clwwlknbp0  41192  clwwlknp  41195  isclwwlksng  41196  clwwlkclwwlkn  41199  clwwnisshclwwsn  41237  clwlksfclwwlk  41269  clwlksfoclwwlk  41270  av-extwwlkfablem2  41510