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Theorem eupthi 29951
Description: Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
Hypothesis
Ref Expression
eupths.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
eupthi (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))

Proof of Theorem eupthi
StepHypRef Expression
1 eupths.i . . 3 𝐼 = (iEdg‘𝐺)
21iseupthf1o 29950 . 2 (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))
32biimpi 215 1 (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533   class class class wbr 5139  dom cdm 5667  1-1-ontowf1o 6533  cfv 6534  (class class class)co 7402  0cc0 11107  ..^cfzo 13628  chash 14291  iEdgciedg 28751  Walkscwlks 29348  EulerPathsceupth 29945
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 5290  ax-nul 5297  ax-pr 5418
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 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-wlks 29351  df-trls 29444  df-eupth 29946
This theorem is referenced by:  eupthf1o  29952  eupthseg  29954  eupthcl  29958  eupthp1  29964
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