MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cycliswlk Structured version   Visualization version   GIF version

Theorem cycliswlk 28788
Description: A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 31-Jan-2021.)
Assertion
Ref Expression
cycliswlk (๐น(Cyclesโ€˜๐บ)๐‘ƒ โ†’ ๐น(Walksโ€˜๐บ)๐‘ƒ)

Proof of Theorem cycliswlk
StepHypRef Expression
1 cyclispth 28787 . 2 (๐น(Cyclesโ€˜๐บ)๐‘ƒ โ†’ ๐น(Pathsโ€˜๐บ)๐‘ƒ)
2 pthiswlk 28717 . 2 (๐น(Pathsโ€˜๐บ)๐‘ƒ โ†’ ๐น(Walksโ€˜๐บ)๐‘ƒ)
31, 2syl 17 1 (๐น(Cyclesโ€˜๐บ)๐‘ƒ โ†’ ๐น(Walksโ€˜๐บ)๐‘ƒ)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   class class class wbr 5106  โ€˜cfv 6497  Walkscwlks 28586  Pathscpths 28702  Cyclesccycls 28775
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 5257  ax-nul 5264  ax-pr 5385
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-wlks 28589  df-trls 28682  df-pths 28706  df-cycls 28777
This theorem is referenced by:  lfgrn1cycl  28792  usgrgt2cycl  33781  usgrcyclgt2v  33782  acycgrcycl  33798  acycgr0v  33799  acycgr1v  33800  prclisacycgr  33802
  Copyright terms: Public domain W3C validator