Theorem pthiswlk 29688

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 29686	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
2		trliswlk 29659	. 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 5095  ‘cfv 6486  Walkscwlks 29560  Trailsctrls 29652  Pathscpths 29673

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-wlks 29563  df-trls 29654  df-pths 29677

This theorem is referenced by:  spthiswlk  29689  pthdadjvtx  29691  2pthnloop  29694  upgr2pthnlp  29695  pthonpth  29711  cycliswlk  29761  cyclnumvtx  29763  wspthsnonn0vne  29880  upgr3v3e3cycl  30142  upgr4cycl4dv4e  30147  pthhashvtx  35100  spthcycl  35101  loop1cycl  35109  upgrimpthslem2  47893  upgrimpths  47894  cycl3grtrilem  47931  cycl3grtri  47932