Theorem pthiswlk 29810

Description: A path is a walk (in an undirected graph). (Contributed by AV, 6-Feb-2021.)

Assertion

Ref	Expression
pthiswlk	⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Proof of Theorem pthiswlk

Step	Hyp	Ref	Expression
1		pthistrl 29808	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
2		trliswlk 29781	. 2 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3	1, 2	syl 17	1 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Syntax hints:   → wi 4   class class class wbr 5100  ‘cfv 6500  Walkscwlks 29682  Trailsctrls 29774  Pathscpths 29795

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-wlks 29685  df-trls 29776  df-pths 29799

This theorem is referenced by:  spthiswlk  29811  pthdadjvtx  29813  2pthnloop  29816  upgr2pthnlp  29817  pthonpth  29833  cycliswlk  29883  cyclnumvtx  29885  wspthsnonn0vne  30002  upgr3v3e3cycl  30267  upgr4cycl4dv4e  30272  pthhashvtx  35341  spthcycl  35342  loop1cycl  35350  upgrimpthslem2  48262  upgrimpths  48263  cycl3grtrilem  48300  cycl3grtri  48301