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Theorem cycliswlk 29628
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 29627 . 2 (๐น(Cyclesโ€˜๐บ)๐‘ƒ โ†’ ๐น(Pathsโ€˜๐บ)๐‘ƒ)
2 pthiswlk 29557 . 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 5141  โ€˜cfv 6541  Walkscwlks 29426  Pathscpths 29542  Cyclesccycls 29615
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 2696  ax-sep 5292  ax-nul 5299  ax-pr 5421
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6493  df-fun 6543  df-fv 6549  df-ov 7417  df-wlks 29429  df-trls 29522  df-pths 29546  df-cycls 29617
This theorem is referenced by:  lfgrn1cycl  29632  usgrgt2cycl  34769  usgrcyclgt2v  34770  acycgrcycl  34786  acycgr0v  34787  acycgr1v  34788  prclisacycgr  34790
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