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Theorem cycliswlk 29044
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 29043 . 2 (๐น(Cyclesโ€˜๐บ)๐‘ƒ โ†’ ๐น(Pathsโ€˜๐บ)๐‘ƒ)
2 pthiswlk 28973 . 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 5147  โ€˜cfv 6540  Walkscwlks 28842  Pathscpths 28958  Cyclesccycls 29031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-wlks 28845  df-trls 28938  df-pths 28962  df-cycls 29033
This theorem is referenced by:  lfgrn1cycl  29048  usgrgt2cycl  34109  usgrcyclgt2v  34110  acycgrcycl  34126  acycgr0v  34127  acycgr1v  34128  prclisacycgr  34130
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