Theorem pthistrl 29758

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)𝑃 → 𝐹(Trails‘𝐺)𝑃)

Proof of Theorem pthistrl

Step	Hyp	Ref	Expression
1		ispth 29756	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
2	1	simp1bi 1144	1 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)

Syntax hints:   → wi 4   = wceq 1537   ∩ cin 3962  ∅c0 4339  {cpr 4633   class class class wbr 5148  ^◡ccnv 5688   ↾ cres 5691   “ cima 5692  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-ne 2939  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-ov 7434  df-trls 29725  df-pths 29749

This theorem is referenced by:  pthiswlk  29760  pthonpth  29781  isspthonpth  29782  usgr2trlspth  29794  usgr2pthspth  29795  cycliscrct  29832  spthcycl  35114