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Theorem relpths 29411
Description: The set (Pathsβ€˜πΊ) 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 (Pathsβ€˜πΊ)

Proof of Theorem relpths
Dummy variables 𝑓 𝑔 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pths 29407 . 2 Paths = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜π‘”)𝑝 ∧ Fun β—‘(𝑝 β†Ύ (1..^(β™―β€˜π‘“))) ∧ ((𝑝 β€œ {0, (β™―β€˜π‘“)}) ∩ (𝑝 β€œ (1..^(β™―β€˜π‘“)))) = βˆ…)})
21relmptopab 7660 1 Rel (Pathsβ€˜πΊ)
Colors of variables: wff setvar class
Syntax hints:   ∧ w3a 1086   = wceq 1540  Vcvv 3473   ∩ cin 3947  βˆ…c0 4322  {cpr 4630   class class class wbr 5148  β—‘ccnv 5675   β†Ύ cres 5678   β€œ cima 5679  Rel wrel 5681  Fun wfun 6537  β€˜cfv 6543  (class class class)co 7412  0cc0 11116  1c1 11117  ..^cfzo 13634  β™―chash 14297  Trailsctrls 29381  Pathscpths 29403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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-pths 29407
This theorem is referenced by:  iscycl  29482  cyclnspth  29491
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