Theorem trliswlk 30286

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 30285	. 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 5102  ^◡ccnv 5649  Fun wfun 6533  ‘cfv 6539  Walkscwlks 30183  Trailsctrls 30279

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fv 6547  df-wlks 30186  df-trls 30281

This theorem is used by:  trlreslem  30288  trlres  30289  trlontrl  30299  pthiswlk  30316  pthdivtx  30318  dfpth2  30320  pthdifv  30322  spthdifv  30325  spthdep  30326  pthdepisspth  30327  usgr2trlspth  30353  crctisclwlk  30387  crctiswlk  30389  crctcshlem3  30414  crctcshwlk  30417  eupthiswlk  30819  eupthres  30822  trlsegvdeglem1  30827  eucrctshift  30850  upgrimtrlslem1  49096  upgrimtrlslem2  49097  upgrimtrls  49098