Theorem clwlkiswlk 29632

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 29631	. 2 ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2	1	simplbi 496	1 ⊢ (𝐹(ClWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Syntax hints:   → wi 4   = wceq 1533   class class class wbr 5143  ‘cfv 6543  0cc0 11138  ♯chash 14321  Walkscwlks 29454  ClWalkscclwlks 29628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fv 6551  df-wlks 29457  df-clwlks 29629

This theorem is referenced by:  clwlkclwwlkfolem  29861