Theorem relwlk 29572

Description: The set (Walks‘𝐺) of all walks on 𝐺 is a set of pairs by our definition of a walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 19-Feb-2021.)

Assertion

Ref	Expression
relwlk	⊢ Rel (Walks‘𝐺)

Proof of Theorem relwlk

Dummy variables 𝑓 𝑔 𝑘 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wlks 29545	. 2 ⊢ Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))})
2	1	relmptopab 7665	1 ⊢ Rel (Walks‘𝐺)

Syntax hints:  if-wif 1062   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  Vcvv 3463   ⊆ wss 3931  {csn 4606  {cpr 4608  dom cdm 5665  Rel wrel 5670  ⟶wf 6537  ‘cfv 6541  (class class class)co 7413  0cc0 11137  1c1 11138   + caddc 11140  ...cfz 13529  ..^cfzo 13676  ♯chash 14351  Word cword 14534  Vtxcvtx 28941  iEdgciedg 28942  Walkscwlks 29542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fv 6549  df-wlks 29545

This theorem is referenced by:  wlkop  29574  istrl  29642  isclwlk  29721  usgrgt2cycl  35094