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Theorem iseupthf1o 30221
Description: The property "𝐹, 𝑃 is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)
Hypothesis
Ref Expression
eupths.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
iseupthf1o (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))

Proof of Theorem iseupthf1o
StepHypRef Expression
1 eupths.i . . 3 𝐼 = (iEdg‘𝐺)
21iseupth 30220 . 2 (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
3 istrl 29714 . . . 4 (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹))
43anbi1i 624 . . 3 ((𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ ((𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹) ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
5 anass 468 . . 3 (((𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹) ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (Fun 𝐹𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)))
6 ancom 460 . . . 4 ((Fun 𝐹𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun 𝐹))
76anbi2i 623 . . 3 ((𝐹(Walks‘𝐺)𝑃 ∧ (Fun 𝐹𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun 𝐹)))
84, 5, 73bitri 297 . 2 ((𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun 𝐹)))
9 dff1o3 6854 . . . 4 (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun 𝐹))
109bicomi 224 . . 3 ((𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun 𝐹) ↔ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)
1110anbi2i 623 . 2 ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun 𝐹)) ↔ (𝐹(Walks‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))
122, 8, 113bitri 297 1 (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540   class class class wbr 5143  ccnv 5684  dom cdm 5685  Fun wfun 6555  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  0cc0 11155  ..^cfzo 13694  chash 14369  iEdgciedg 29014  Walkscwlks 29614  Trailsctrls 29708  EulerPathsceupth 30216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-wlks 29617  df-trls 29710  df-eupth 30217
This theorem is referenced by:  eupthi  30222  upgriseupth  30226  eupth0  30233  eupthres  30234  eupthp1  30235
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