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Theorem relwlk 29138
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 29111 . 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 7658 1 Rel (Walksβ€˜πΊ)
Colors of variables: wff setvar class
Syntax hints:  if-wif 1061   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βŠ† wss 3948  {csn 4628  {cpr 4630  dom cdm 5676  Rel wrel 5681  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115  ...cfz 13488  ..^cfzo 13631  β™―chash 14294  Word cword 14468  Vtxcvtx 28511  iEdgciedg 28512  Walkscwlks 29108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-wlks 29111
This theorem is referenced by:  wlkop  29140  istrl  29208  isclwlk  29285  usgrgt2cycl  34407
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