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Theorem clwlkwlk 29609
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 (π‘Š ∈ (ClWalksβ€˜πΊ) β†’ π‘Š ∈ (Walksβ€˜πΊ))

Proof of Theorem clwlkwlk
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elopabran 5568 . 2 (π‘Š ∈ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)))} β†’ π‘Š ∈ (Walksβ€˜πΊ))
2 clwlks 29606 . 2 (ClWalksβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)))}
31, 2eleq2s 2847 1 (π‘Š ∈ (ClWalksβ€˜πΊ) β†’ π‘Š ∈ (Walksβ€˜πΊ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   class class class wbr 5152  {copab 5214  β€˜cfv 6553  0cc0 11146  β™―chash 14329  Walkscwlks 29430  ClWalkscclwlks 29604
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-clwlks 29605
This theorem is referenced by:  clwlkswks  29610  upgrclwlkcompim  29615  clwlkcompbp  29616  clwlkclwwlkflem  29834  clwlknf1oclwwlknlem1  29911  clwlknf1oclwwlkn  29914  clwwlknonclwlknonf1o  30192  dlwwlknondlwlknonf1olem1  30194  wlkl0  30197
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