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Theorem clwlksfclwwlk1hashn 41266
 Description: The size of the first component of a closed walk. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 2-May-2021.)
Hypotheses
Ref Expression
clwlksfclwwlk.1 𝐴 = (1st𝑐)
clwlksfclwwlk.2 𝐵 = (2nd𝑐)
clwlksfclwwlk.c 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
clwlksfclwwlk.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
Assertion
Ref Expression
clwlksfclwwlk1hashn (𝑊𝐶 → (#‘(1st𝑊)) = 𝑁)
Distinct variable groups:   𝐺,𝑐   𝑁,𝑐   𝑊,𝑐
Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)   𝐶(𝑐)   𝐹(𝑐)

Proof of Theorem clwlksfclwwlk1hashn
StepHypRef Expression
1 clwlksfclwwlk.1 . . . . . 6 𝐴 = (1st𝑐)
21fveq2i 6106 . . . . 5 (#‘𝐴) = (#‘(1st𝑐))
32eqeq1i 2615 . . . 4 ((#‘𝐴) = 𝑁 ↔ (#‘(1st𝑐)) = 𝑁)
4 fveq2 6103 . . . . . 6 (𝑐 = 𝑊 → (1st𝑐) = (1st𝑊))
54fveq2d 6107 . . . . 5 (𝑐 = 𝑊 → (#‘(1st𝑐)) = (#‘(1st𝑊)))
65eqeq1d 2612 . . . 4 (𝑐 = 𝑊 → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘(1st𝑊)) = 𝑁))
73, 6syl5bb 271 . . 3 (𝑐 = 𝑊 → ((#‘𝐴) = 𝑁 ↔ (#‘(1st𝑊)) = 𝑁))
8 clwlksfclwwlk.c . . 3 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
97, 8elrab2 3333 . 2 (𝑊𝐶 ↔ (𝑊 ∈ (ClWalkS‘𝐺) ∧ (#‘(1st𝑊)) = 𝑁))
109simprbi 479 1 (𝑊𝐶 → (#‘(1st𝑊)) = 𝑁)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900  ⟨cop 4131   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  #chash 12979   substr csubstr 13150  ClWalkScclwlks 40976 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  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-iota 5768  df-fv 5812 This theorem is referenced by:  clwlksf1clwwlklem  41275  clwlksf1clwwlk  41276
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