Theorem cycliscrct 30089

Description: A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)

Assertion

Ref	Expression
cycliscrct	⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Circuits‘𝐺)𝑃)

Proof of Theorem cycliscrct

Step	Hyp	Ref	Expression
1		pthistrl 30013	. . 3 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
2	1	anim1i 626	. 2 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
3		iscycl 30081	. 2 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
4		iscrct 30080	. 2 ⊢ (𝐹(Circuits‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
5	2, 3, 4	3imtr4i 295	1 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Circuits‘𝐺)𝑃)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   class class class wbr 5113  ‘cfv 6537  0cc0 11100  ♯chash 14366  Trailsctrls 29979  Pathscpths 30000  Circuitsccrcts 30074  Cyclesccycls 30075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-trls 29981  df-pths 30004  df-crcts 30076  df-cycls 30077

This theorem is referenced by:  usgrn2cycl  30099