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Theorem pthiswlk 29760
Description: A path is a walk (in an undirected graph). (Contributed by AV, 6-Feb-2021.)
Assertion
Ref Expression
pthiswlk (𝐹(Paths‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)

Proof of Theorem pthiswlk
StepHypRef Expression
1 pthistrl 29758 . 2 (𝐹(Paths‘𝐺)𝑃𝐹(Trails‘𝐺)𝑃)
2 trliswlk 29730 . 2 (𝐹(Trails‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)
31, 2syl 17 1 (𝐹(Paths‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 5148  cfv 6563  Walkscwlks 29629  Trailsctrls 29723  Pathscpths 29745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-wlks 29632  df-trls 29725  df-pths 29749
This theorem is referenced by:  spthiswlk  29761  pthdadjvtx  29763  2pthnloop  29764  upgr2pthnlp  29765  pthonpth  29781  cycliswlk  29831  wspthsnonn0vne  29947  upgr3v3e3cycl  30209  upgr4cycl4dv4e  30214  pthhashvtx  35112  spthcycl  35114  loop1cycl  35122
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