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Theorem relpths 26672
 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 26668 . 2 Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
21relmptopab 6925 1 Rel (Paths‘𝐺)
 Colors of variables: wff setvar class Syntax hints:   ∧ w3a 1054   = wceq 1523  Vcvv 3231   ∩ cin 3606  ∅c0 3948  {cpr 4212   class class class wbr 4685  ◡ccnv 5142   ↾ cres 5145   “ cima 5146  Rel wrel 5148  Fun wfun 5920  ‘cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975  ..^cfzo 12504  #chash 13157  Trailsctrls 26643  Pathscpths 26664 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-pths 26668 This theorem is referenced by:  iscycl  26742  cyclnspth  26751
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