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Theorem eupthistrl 29973
Description: An Eulerian path is a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.) (Revised by AV, 18-Feb-2021.)
Assertion
Ref Expression
eupthistrl (𝐹(EulerPaths‘𝐺)𝑃𝐹(Trails‘𝐺)𝑃)

Proof of Theorem eupthistrl
StepHypRef Expression
1 eqid 2726 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
21iseupth 29963 . 2 (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–onto→dom (iEdg‘𝐺)))
32simplbi 497 1 (𝐹(EulerPaths‘𝐺)𝑃𝐹(Trails‘𝐺)𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 5141  dom cdm 5669  ontowfo 6535  cfv 6537  (class class class)co 7405  0cc0 11112  ..^cfzo 13633  chash 14295  iEdgciedg 28765  Trailsctrls 29456  EulerPathsceupth 29959
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-fo 6543  df-fv 6545  df-ov 7408  df-trls 29458  df-eupth 29960
This theorem is referenced by:  eupthiswlk  29974  eupthres  29977  eupth2eucrct  29979  eupth2lem3  29998
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