Theorem crctistrl 29766

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 29763	. 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 5089  ‘cfv 6477  0cc0 10998  ♯chash 14229  Trailsctrls 29660  Circuitsccrcts 29755

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fv 6485  df-trls 29662  df-crcts 29757

This theorem is referenced by:  crctiswlk  29767  crctcshlem3  29790  crctcshwlk  29793  crctcshtrl  29794