Theorem pthiswlk 29707

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 29705	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
2		trliswlk 29677	. 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 5119  ‘cfv 6531  Walkscwlks 29576  Trailsctrls 29670  Pathscpths 29692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-wlks 29579  df-trls 29672  df-pths 29696

This theorem is referenced by:  spthiswlk  29708  pthdadjvtx  29710  2pthnloop  29713  upgr2pthnlp  29714  pthonpth  29730  cycliswlk  29780  cyclnumvtx  29782  wspthsnonn0vne  29899  upgr3v3e3cycl  30161  upgr4cycl4dv4e  30166  pthhashvtx  35150  spthcycl  35151  loop1cycl  35159  upgrimpthslem2  47921  upgrimpths  47922  cycl3grtrilem  47958  cycl3grtri  47959