Theorem clwlkwlk 29541

Description: Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)

Assertion

Ref	Expression
clwlkwlk	⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺))

Proof of Theorem clwlkwlk

Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elopabran 5555	. 2 ⊢ (𝑊 ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))} → 𝑊 ∈ (Walks‘𝐺))
2		clwlks 29538	. 2 ⊢ (ClWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
3	1, 2	eleq2s 2845	1 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  {copab 5203  ‘cfv 6537  0cc0 11112  ♯chash 14295  Walkscwlks 29362  ClWalkscclwlks 29536

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-clwlks 29537

This theorem is referenced by:  clwlkswks  29542  upgrclwlkcompim  29547  clwlkcompbp  29548  clwlkclwwlkflem  29766  clwlknf1oclwwlknlem1  29843  clwlknf1oclwwlkn  29846  clwwlknonclwlknonf1o  30124  dlwwlknondlwlknonf1olem1  30126  wlkl0  30129