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Theorem clwlks 29029
Description: The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) (Revised by AV, 29-Oct-2021.)
Assertion
Ref Expression
clwlks (ClWalksβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)))}
Distinct variable group:   𝑓,𝐺,𝑝

Proof of Theorem clwlks
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 biidd 262 . 2 (𝑔 = 𝐺 β†’ ((π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)) ↔ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“))))
2 df-clwlks 29028 . 2 ClWalks = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜π‘”)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)))})
31, 2fvmptopab 7463 1 (ClWalksβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)))}
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 397   = wceq 1542   class class class wbr 5149  {copab 5211  β€˜cfv 6544  0cc0 11110  β™―chash 14290  Walkscwlks 28853  ClWalkscclwlks 29027
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-clwlks 29028
This theorem is referenced by:  isclwlk  29030  clwlkwlk  29032  clwlkcompim  29037
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