Theorem iscrct 30392

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.

Step	Hyp	Ref	Expression
1		crcts 30390	. 2 ⊢ (Circuits‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
2		fveq1 6895	. . . 4 ⊢ (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
3	2	adantl 487	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑝‘0) = (𝑃‘0))
4		simpr 490	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
5		fveq2 6896	. . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
6	5	adantr 486	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (♯‘𝑓) = (♯‘𝐹))
7	4, 6	fveq12d 6903	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑝‘(♯‘𝑓)) = (𝑃‘(♯‘𝐹)))
8	3, 7	eqeq12d 2779	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
9		reltrls 30292	. 2 ⊢ Rel (Trails‘𝐺)
10	1, 8, 9	brfvopabrbr 7001	1 ⊢ (𝐹(Circuits‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   class class class wbr 5106  ‘cfv 6547  0cc0 11213  ♯chash 14487  Trailsctrls 30288  Circuitsccrcts 30386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fv 6555  df-trls 30290  df-crcts 30388

This theorem is used by:  crctprop  30394  cycliscrct  30402  crctcsh  30428  0crct  30739  eupth2eucrct  30833