Theorem clwlkiswlk 29863

Description: A closed walk is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)

Assertion

Ref	Expression
clwlkiswlk	⊢ (𝐹(ClWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Proof of Theorem clwlkiswlk

Step	Hyp	Ref	Expression
1		isclwlk 29862	. 2 ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2	1	simplbi 497	1 ⊢ (𝐹(ClWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Syntax hints:   → wi 4   = wceq 1543   class class class wbr 5075  ‘cfv 6488  0cc0 11032  ♯chash 14286  Walkscwlks 29686  ClWalkscclwlks 29859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-10 2148  ax-11 2164  ax-12 2185  ax-ext 2708  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-nf 1787  df-sb 2070  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2932  df-ral 3051  df-rex 3061  df-rab 3389  df-v 3430  df-sbc 3727  df-csb 3835  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fv 6496  df-wlks 29689  df-clwlks 29860

This theorem is referenced by:  clwlkclwwlkfolem  30098