Theorem erclwwlkrel 30109

Description: ∼ is a relation. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.)

Hypothesis

Ref	Expression
erclwwlk.r	⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}

Assertion

Ref	Expression
erclwwlkrel	⊢ Rel ∼

Proof of Theorem erclwwlkrel

Step	Hyp	Ref	Expression
1		erclwwlk.r	. 2 ⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
2	1	relopabi 5768	1 ⊢ Rel ∼

Syntax hints:   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∃wrex 3065  {copab 5137  Rel wrel 5626  ‘cfv 6489  (class class class)co 7360  0cc0 11033  ...cfz 13456  ♯chash 14287   cyclShift ccsh 14745  ClWWalkscclwwlk 30073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5138  df-xp 5627  df-rel 5628

This theorem is referenced by:  erclwwlk  30115