Theorem clwlkiswlk 30394

Description: A closed walk is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)

Assertion

Ref	Expression
clwlkiswlk	⊢ (𝐹(ClWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Proof of Theorem clwlkiswlk

Step	Hyp	Ref	Expression
1		isclwlk 30393	. 2 ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2	1	simplbi 502	1 ⊢ (𝐹(ClWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   class class class wbr 5106  ‘cfv 6561  0cc0 11230  ♯chash 14504  Walkscwlks 30210  ClWalkscclwlks 30390

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fv 6569  df-wlks 30213  df-clwlks 30391

This theorem is used by:  clwlkclwwlkfolem  30631