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Theorem relpths 29753
Description: The set (Paths‘𝐺) of all paths on 𝐺 is a set of pairs by our definition of a path, and so is a relation. (Contributed by AV, 30-Oct-2021.)
Assertion
Ref Expression
relpths Rel (Paths‘𝐺)

Proof of Theorem relpths
Dummy variables 𝑓 𝑔 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pths 29749 . 2 Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
21relmptopab 7683 1 Rel (Paths‘𝐺)
Colors of variables: wff setvar class
Syntax hints:  w3a 1086   = wceq 1537  Vcvv 3478  cin 3962  c0 4339  {cpr 4633   class class class wbr 5148  ccnv 5688  cres 5691  cima 5692  Rel wrel 5694  Fun wfun 6557  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  ..^cfzo 13691  chash 14366  Trailsctrls 29723  Pathscpths 29745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-pths 29749
This theorem is referenced by:  iscycl  29824  cyclnspth  29833
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