Theorem clwlks 29498

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	⊢ (ClWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}

Distinct variable group:   𝑓,𝐺,𝑝

Proof of Theorem clwlks

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		biidd 262	. 2 ⊢ (𝑔 = 𝐺 → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑝‘0) = (𝑝‘(♯‘𝑓))))
2		df-clwlks 29497	. 2 ⊢ ClWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
3	1, 2	fvmptopab 7455	1 ⊢ (ClWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}

Syntax hints:   ∧ wa 395   = wceq 1533   class class class wbr 5138  {copab 5200  ‘cfv 6533  0cc0 11106  ♯chash 14287  Walkscwlks 29322  ClWalkscclwlks 29496

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 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6485  df-fun 6535  df-fv 6541  df-clwlks 29497

This theorem is referenced by:  isclwlk  29499  clwlkwlk  29501  clwlkcompim  29506