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Theorem relwlk 26815
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.
StepHypRef Expression
1 df-wlks 26789 . 2 Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))))})
21relmptopab 7083 1 Rel (Walks‘𝐺)
Colors of variables: wff setvar class
Syntax hints:  if-wif 1085  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  wss 3734  {csn 4336  {cpr 4338  dom cdm 5279  Rel wrel 5284  wf 6066  cfv 6070  (class class class)co 6844  0cc0 10191  1c1 10192   + caddc 10194  ...cfz 12536  ..^cfzo 12676  chash 13324  Word cword 13489  Vtxcvtx 26168  iEdgciedg 26169  Walkscwlks 26786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fv 6078  df-wlks 26789
This theorem is referenced by:  wlkop  26817  istrl  26888  isclwlk  26964
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