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

Theorem iscycl 29823
Description: Sufficient and necessary conditions for a pair of functions to be a cycle (in an undirected graph): A pair of function "is" (represents) a cycle iff it is a closed path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.)
Assertion
Ref Expression
iscycl (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Proof of Theorem iscycl
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cycls 29821 . 2 (Cycles‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
2 fveq1 6905 . . . 4 (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
32adantl 481 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝‘0) = (𝑃‘0))
4 simpr 484 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → 𝑝 = 𝑃)
5 fveq2 6906 . . . . 5 (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
65adantr 480 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (♯‘𝑓) = (♯‘𝐹))
74, 6fveq12d 6913 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝‘(♯‘𝑓)) = (𝑃‘(♯‘𝐹)))
83, 7eqeq12d 2750 . 2 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
9 relpths 29752 . 2 Rel (Paths‘𝐺)
101, 8, 9brfvopabrbr 7012 1 (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536   class class class wbr 5147  cfv 6562  0cc0 11152  chash 14365  Pathscpths 29744  Cyclesccycls 29817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570  df-pths 29748  df-cycls 29819
This theorem is referenced by:  cyclprop  29825  cycliscrct  29831  cyclnspth  29832  cyclispthon  29833  0cycl  30162  lp1cycl  30180  3cycld  30206  pthisspthorcycl  35112  spthcycl  35113  subgrcycl  35119  2cycld  35122
  Copyright terms: Public domain W3C validator