Theorem clwlkwlk 29796

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 5566	. 2 ⊢ (𝑊 ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))} → 𝑊 ∈ (Walks‘𝐺))
2		clwlks 29793	. 2 ⊢ (ClWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
3	1, 2	eleq2s 2858	1 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   class class class wbr 5142  {copab 5204  ‘cfv 6560  0cc0 11156  ♯chash 14370  Walkscwlks 29615  ClWalkscclwlks 29791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-clwlks 29792

This theorem is referenced by:  clwlkswks  29797  upgrclwlkcompim  29802  clwlkcompbp  29803  clwlkclwwlkflem  30024  clwlknf1oclwwlknlem1  30101  clwlknf1oclwwlkn  30104  clwwlknonclwlknonf1o  30382  dlwwlknondlwlknonf1olem1  30384  wlkl0  30387