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Theorem trliswlk 28474
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)𝑃𝐹(Walks‘𝐺)𝑃)

Proof of Theorem trliswlk
StepHypRef Expression
1 istrl 28473 . 2 (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹))
21simplbi 498 1 (𝐹(Trails‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 5103  ccnv 5630  Fun wfun 6487  cfv 6493  Walkscwlks 28373  Trailsctrls 28467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fv 6501  df-wlks 28376  df-trls 28469
This theorem is referenced by:  trlreslem  28476  trlres  28477  trlontrl  28488  pthiswlk  28504  pthdivtx  28506  spthdifv  28510  spthdep  28511  pthdepisspth  28512  usgr2trlspth  28538  crctisclwlk  28571  crctiswlk  28573  crctcshlem3  28593  crctcshwlk  28596  eupthiswlk  28985  eupthres  28988  trlsegvdeglem1  28993  eucrctshift  29016
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