Theorem crctistrl 29775

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

Assertion

Ref	Expression
crctistrl	⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)

Proof of Theorem crctistrl

Step	Hyp	Ref	Expression
1		crctprop 29772	. 2 ⊢ (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2	1	simpld 494	1 ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)

Syntax hints:   → wi 4   = wceq 1541   class class class wbr 5093  ‘cfv 6486  0cc0 11013  ♯chash 14239  Trailsctrls 29669  Circuitsccrcts 29764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fv 6494  df-trls 29671  df-crcts 29766

This theorem is referenced by:  crctiswlk  29776  crctcshlem3  29799  crctcshwlk  29802  crctcshtrl  29803