Theorem cycliswlk 29820

Description: A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 31-Jan-2021.)

Assertion

Ref	Expression
cycliswlk	⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Proof of Theorem cycliswlk

Step	Hyp	Ref	Expression
1		cyclispth 29819	. 2 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃)
2		pthiswlk 29747	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3	1, 2	syl 17	1 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Syntax hints:   → wi 4   class class class wbr 5096  ‘cfv 6490  Walkscwlks 29619  Pathscpths 29732  Cyclesccycls 29807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-wlks 29622  df-trls 29713  df-pths 29736  df-cycls 29809

This theorem is referenced by:  lfgrn1cycl  29827  usgrgt2cycl  35273  usgrcyclgt2v  35274  acycgrcycl  35290  acycgr0v  35291  acycgr1v  35292  prclisacycgr  35294  upgrimcycls  48099  cycldlenngric  48116