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Theorem crctistrl 29487
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 (๐น(Circuitsโ€˜๐บ)๐‘ƒ โ†’ ๐น(Trailsโ€˜๐บ)๐‘ƒ)

Proof of Theorem crctistrl
StepHypRef Expression
1 crctprop 29484 . 2 (๐น(Circuitsโ€˜๐บ)๐‘ƒ โ†’ (๐น(Trailsโ€˜๐บ)๐‘ƒ โˆง (๐‘ƒโ€˜0) = (๐‘ƒโ€˜(โ™ฏโ€˜๐น))))
21simpld 494 1 (๐น(Circuitsโ€˜๐บ)๐‘ƒ โ†’ ๐น(Trailsโ€˜๐บ)๐‘ƒ)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   = wceq 1533   class class class wbr 5138  โ€˜cfv 6533  0cc0 11105  โ™ฏchash 14286  Trailsctrls 29382  Circuitsccrcts 29476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fv 6541  df-trls 29384  df-crcts 29478
This theorem is referenced by:  crctiswlk  29488  crctcshlem3  29508  crctcshwlk  29511  crctcshtrl  29512
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