Theorem trliswlk 29675

Description: A trail is a walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 29-Oct-2021.)

Assertion

Ref	Expression
trliswlk	⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Proof of Theorem trliswlk

Step	Hyp	Ref	Expression
1		istrl 29674	. 2 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))
2	1	simplbi 497	1 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Syntax hints:   → wi 4   class class class wbr 5091  ^◡ccnv 5615  Fun wfun 6475  ‘cfv 6481  Walkscwlks 29576  Trailsctrls 29668

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 5234  ax-nul 5244  ax-pr 5370

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 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fv 6489  df-wlks 29579  df-trls 29670

This theorem is referenced by:  trlreslem  29677  trlres  29678  trlontrl  29688  pthiswlk  29704  pthdivtx  29706  dfpth2  29708  pthdifv  29709  spthdifv  29712  spthdep  29713  pthdepisspth  29714  usgr2trlspth  29740  crctisclwlk  29773  crctiswlk  29775  crctcshlem3  29798  crctcshwlk  29801  eupthiswlk  30190  eupthres  30193  trlsegvdeglem1  30198  eucrctshift  30221  upgrimtrlslem1  47941  upgrimtrlslem2  47942  upgrimtrls  47943