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Theorem iscycl 29697
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 29695 . 2 (Cycles‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
2 fveq1 6895 . . . 4 (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
32adantl 480 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝‘0) = (𝑃‘0))
4 simpr 483 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → 𝑝 = 𝑃)
5 fveq2 6896 . . . . 5 (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
65adantr 479 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (♯‘𝑓) = (♯‘𝐹))
74, 6fveq12d 6903 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝‘(♯‘𝑓)) = (𝑃‘(♯‘𝐹)))
83, 7eqeq12d 2741 . 2 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
9 relpths 29626 . 2 Rel (Paths‘𝐺)
101, 8, 9brfvopabrbr 7001 1 (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533   class class class wbr 5149  cfv 6549  0cc0 11145  chash 14333  Pathscpths 29618  Cyclesccycls 29691
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 5300  ax-nul 5307  ax-pr 5429
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 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557  df-pths 29622  df-cycls 29693
This theorem is referenced by:  cyclprop  29699  cycliscrct  29705  cyclnspth  29706  cyclispthon  29707  0cycl  30036  lp1cycl  30054  3cycld  30080  pthisspthorcycl  34889  spthcycl  34890  subgrcycl  34896  2cycld  34899
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