Theorem cycliscrct 29779

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 29703	. . 3 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
2	1	anim1i 615	. 2 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
3		iscycl 29771	. 2 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
4		iscrct 29770	. 2 ⊢ (𝐹(Circuits‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
5	2, 3, 4	3imtr4i 292	1 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Circuits‘𝐺)𝑃)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   class class class wbr 5102  ‘cfv 6499  0cc0 11044  ♯chash 14271  Trailsctrls 29669  Pathscpths 29690  Circuitsccrcts 29764  Cyclesccycls 29765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-trls 29671  df-pths 29694  df-crcts 29766  df-cycls 29767

This theorem is referenced by:  usgrn2cycl  29789