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Theorem relwlk 25784
Description: The walks (in an undirected simple graph) are (ordered) pairs. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
Assertion
Ref Expression
relwlk Rel (𝑉 Walks 𝐸)

Proof of Theorem relwlk
Dummy variables 𝑒 𝑓 𝑘 𝑝 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wlk 25774 . 2 Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
21relmpt2opab 7021 1 Rel (𝑉 Walks 𝐸)
Colors of variables: wff setvar class
Syntax hints:  w3a 1030   = wceq 1474  wcel 1938  wral 2800  Vcvv 3077  {cpr 4030  dom cdm 4932  Rel wrel 4937  wf 5685  cfv 5689  (class class class)co 6426  0cc0 9691  1c1 9692   + caddc 9694  ...cfz 12065  ..^cfzo 12202  #chash 12847  Word cword 13005   Walks cwalk 25764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-1st 6934  df-2nd 6935  df-wlk 25774
This theorem is referenced by:  wlkop  25794
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