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Theorem pthistrl 29611
Description: A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
Assertion
Ref Expression
pthistrl (𝐹(Paths‘𝐺)𝑃𝐹(Trails‘𝐺)𝑃)

Proof of Theorem pthistrl
StepHypRef Expression
1 ispth 29609 . 2 (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun (𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
21simp1bi 1142 1 (𝐹(Paths‘𝐺)𝑃𝐹(Trails‘𝐺)𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cin 3943  c0 4322  {cpr 4632   class class class wbr 5149  ccnv 5677  cres 5680  cima 5681  Fun wfun 6543  cfv 6549  (class class class)co 7419  0cc0 11140  1c1 11141  ..^cfzo 13662  chash 14325  Trailsctrls 29576  Pathscpths 29598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-trls 29578  df-pths 29602
This theorem is referenced by:  pthiswlk  29613  pthonpth  29634  isspthonpth  29635  usgr2trlspth  29647  usgr2pthspth  29648  cycliscrct  29685  spthcycl  34867
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