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

Proof of Theorem iscrct
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crcts 29821 . 2 (Circuits‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
2 fveq1 6906 . . . 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 6907 . . . . 5 (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
65adantr 480 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (♯‘𝑓) = (♯‘𝐹))
74, 6fveq12d 6914 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝‘(♯‘𝑓)) = (𝑃‘(♯‘𝐹)))
83, 7eqeq12d 2751 . 2 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
9 reltrls 29727 . 2 Rel (Trails‘𝐺)
101, 8, 9brfvopabrbr 7013 1 (𝐹(Circuits‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537   class class class wbr 5148  cfv 6563  0cc0 11153  chash 14366  Trailsctrls 29723  Circuitsccrcts 29817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-trls 29725  df-crcts 29819
This theorem is referenced by:  crctprop  29825  cycliscrct  29832  crctcsh  29854  0crct  30162  eupth2eucrct  30246
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