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Theorem iscrct 28976
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 28974 . 2 (Circuits‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
2 fveq1 6878 . . . 4 (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
32adantl 482 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝‘0) = (𝑃‘0))
4 simpr 485 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → 𝑝 = 𝑃)
5 fveq2 6879 . . . . 5 (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
65adantr 481 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (♯‘𝑓) = (♯‘𝐹))
74, 6fveq12d 6886 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝‘(♯‘𝑓)) = (𝑃‘(♯‘𝐹)))
83, 7eqeq12d 2748 . 2 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
9 reltrls 28880 . 2 Rel (Trails‘𝐺)
101, 8, 9brfvopabrbr 6982 1 (𝐹(Circuits‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541   class class class wbr 5142  cfv 6533  0cc0 11094  chash 14274  Trailsctrls 28876  Circuitsccrcts 28970
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 5293  ax-nul 5300  ax-pr 5421
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 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fv 6541  df-trls 28878  df-crcts 28972
This theorem is referenced by:  crctprop  28978  cycliscrct  28985  crctcsh  29007  0crct  29315  eupth2eucrct  29399
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