Theorem cycliscrct 30148

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 30072	. . 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 30140	. 2 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
4		iscrct 30139	. 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 1570   class class class wbr 5109  ‘cfv 6536  0cc0 11095  ♯chash 14362  Trailsctrls 30038  Pathscpths 30059  Circuitsccrcts 30133  Cyclesccycls 30134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-trls 30040  df-pths 30063  df-crcts 30135  df-cycls 30136

This theorem is referenced by:  usgrn2cycl  30158