Theorem pthiswlk 30340

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 30338	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
2		trliswlk 30310	. 2 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3	1, 2	syl 18	1 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

This proof depends on syntax axioms:   → wi 4   class class class wbr 5106  ‘cfv 6558  Walkscwlks 30207  Trailsctrls 30303  Pathscpths 30325

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 5252  ax-nul 5263  ax-pr 5395

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 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-wlks 30210  df-trls 30305  df-pths 30329

This theorem is used by:  spthiswlk  30341  pthdadjvtx  30343  pthhashvtx  30345  2pthnloop  30347  upgr2pthnlp  30348  pthonpth  30364  cycliswlk  30416  cyclnumvtx  30418  spthcycl  30422  wspthsnonn0vne  30536  loop1cycl  30774  upgr3v3e3cycl  30811  upgr4cycl4dv4e  30816  upgrimpthslem2  49211  upgrimpths  49212  cycl3grtrilem  49249  cycl3grtri  49250