Theorem cyclispth 29729

Description: A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)

Assertion

Ref	Expression
cyclispth	⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃)

Proof of Theorem cyclispth

Step	Hyp	Ref	Expression
1		cyclprop 29725	. 2 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2	1	simpld 493	1 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃)

Syntax hints:   → wi 4   = wceq 1534   class class class wbr 5144  ‘cfv 6544  0cc0 11147  ♯chash 14340  Pathscpths 29644  Cyclesccycls 29717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fv 6552  df-pths 29648  df-cycls 29719

This theorem is referenced by:  cycliswlk  29730  cyclispthon  29733  usgrcyclgt2v  34970  acycgr1v  34988  pthacycspth  34996