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Theorem relwlk 28883
Description: The set (Walksβ€˜πΊ) 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 (Walksβ€˜πΊ)

Proof of Theorem relwlk
Dummy variables 𝑓 𝑔 π‘˜ 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wlks 28856 . 2 Walks = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom (iEdgβ€˜π‘”) ∧ 𝑝:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜π‘”) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜))))})
21relmptopab 7656 1 Rel (Walksβ€˜πΊ)
Colors of variables: wff setvar class
Syntax hints:  if-wif 1062   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475   βŠ† wss 3949  {csn 4629  {cpr 4631  dom cdm 5677  Rel wrel 5682  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113  ...cfz 13484  ..^cfzo 13627  β™―chash 14290  Word cword 14464  Vtxcvtx 28256  iEdgciedg 28257  Walkscwlks 28853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-wlks 28856
This theorem is referenced by:  wlkop  28885  istrl  28953  isclwlk  29030  usgrgt2cycl  34121
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