Theorem pthistrl 29686

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 29684	. 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 1540   ∩ cin 3904  ∅c0 4286  {cpr 4581   class class class wbr 5095  ^◡ccnv 5622   ↾ cres 5625   “ cima 5626  Fun wfun 6480  ‘cfv 6486  (class class class)co 7353  0cc0 11028  1c1 11029  ..^cfzo 13575  ♯chash 14255  Trailsctrls 29652  Pathscpths 29673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-trls 29654  df-pths 29677

This theorem is referenced by:  pthiswlk  29688  pthonpth  29711  isspthonpth  29712  usgr2trlspth  29724  usgr2pthspth  29725  cycliscrct  29762  spthcycl  35104  upgrimpths  47897  upgrimspths  47898