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Theorem eupthistrl 30060
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 2725 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
21iseupth 30050 . 2 (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–onto→dom (iEdg‘𝐺)))
32simplbi 496 1 (𝐹(EulerPaths‘𝐺)𝑃𝐹(Trails‘𝐺)𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 5144  dom cdm 5673  ontowfo 6541  cfv 6543  (class class class)co 7413  0cc0 11133  ..^cfzo 13654  chash 14316  iEdgciedg 28849  Trailsctrls 29543  EulerPathsceupth 30046
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 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-fo 6549  df-fv 6551  df-ov 7416  df-trls 29545  df-eupth 30047
This theorem is referenced by:  eupthiswlk  30061  eupthres  30064  eupth2eucrct  30066  eupth2lem3  30085
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