Theorem pthistrl 29696

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 29694	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
2	1	simp1bi 1145	1 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)

Syntax hints:   → wi 4   = wceq 1541   ∩ cin 3896  ∅c0 4278  {cpr 4573   class class class wbr 5086  ^◡ccnv 5610   ↾ cres 5613   “ cima 5614  Fun wfun 6470  ‘cfv 6476  (class class class)co 7341  0cc0 11001  1c1 11002  ..^cfzo 13549  ♯chash 14232  Trailsctrls 29662  Pathscpths 29683

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fv 6484  df-ov 7344  df-trls 29664  df-pths 29687

This theorem is referenced by:  pthiswlk  29698  pthonpth  29721  isspthonpth  29722  usgr2trlspth  29734  usgr2pthspth  29735  cycliscrct  29772  spthcycl  35165  upgrimpths  47940  upgrimspths  47941