Theorem relwlk 29590

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 29564	. 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 7603	1 ⊢ Rel (Walks‘𝐺)

Syntax hints:  if-wif 1062   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3438   ⊆ wss 3905  {csn 4579  {cpr 4581  dom cdm 5623  Rel wrel 5628  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  0cc0 11028  1c1 11029   + caddc 11031  ...cfz 13429  ..^cfzo 13576  ♯chash 14256  Word cword 14439  Vtxcvtx 28960  iEdgciedg 28961  Walkscwlks 29561

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 5238  ax-nul 5248  ax-pr 5374

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-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fv 6494  df-wlks 29564

This theorem is referenced by:  wlkop  29592  istrl  29659  isclwlk  29737  usgrgt2cycl  35122