Theorem trliswlk 30313

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 30312	. 2 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))
2	1	simplbi 502	1 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

This proof depends on syntax axioms:   → wi 4   class class class wbr 5106  ^◡ccnv 5655  Fun wfun 6555  ‘cfv 6561  Walkscwlks 30210  Trailsctrls 30306

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fv 6569  df-wlks 30213  df-trls 30308

This theorem is used by:  trlreslem  30315  trlres  30316  trlontrl  30326  pthiswlk  30343  pthdivtx  30345  dfpth2  30347  pthdifv  30349  spthdifv  30352  spthdep  30353  pthdepisspth  30354  usgr2trlspth  30380  crctisclwlk  30414  crctiswlk  30416  crctcshlem3  30441  crctcshwlk  30444  eupthiswlk  30846  eupthres  30849  trlsegvdeglem1  30854  eucrctshift  30877  upgrimtrlslem1  49217  upgrimtrlslem2  49218  upgrimtrls  49219