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Theorem relwlk 28616
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 28589 . 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 7604 1 Rel (Walksβ€˜πΊ)
Colors of variables: wff setvar class
Syntax hints:  if-wif 1062   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444   βŠ† wss 3911  {csn 4587  {cpr 4589  dom cdm 5634  Rel wrel 5639  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  0cc0 11056  1c1 11057   + caddc 11059  ...cfz 13430  ..^cfzo 13573  β™―chash 14236  Word cword 14408  Vtxcvtx 27989  iEdgciedg 27990  Walkscwlks 28586
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 5257  ax-nul 5264  ax-pr 5385
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 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-wlks 28589
This theorem is referenced by:  wlkop  28618  istrl  28686  isclwlk  28763  usgrgt2cycl  33781
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