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Theorem iseupthf1o 28975
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 28974 . 2 (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
3 istrl 28473 . . . 4 (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹))
43anbi1i 625 . . 3 ((𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ ((𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹) ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
5 anass 470 . . 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 462 . . . 4 ((Fun 𝐹𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun 𝐹))
76anbi2i 624 . . 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 6788 . . . 4 (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun 𝐹))
109bicomi 223 . . 3 ((𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun 𝐹) ↔ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)
1110anbi2i 624 . 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 205  wa 397   = wceq 1542   class class class wbr 5104  ccnv 5631  dom cdm 5632  Fun wfun 6488  ontowfo 6492  1-1-ontowf1o 6493  cfv 6494  (class class class)co 7352  0cc0 11010  ..^cfzo 13522  chash 14184  iEdgciedg 27777  Walkscwlks 28373  Trailsctrls 28467  EulerPathsceupth 28970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7355  df-wlks 28376  df-trls 28469  df-eupth 28971
This theorem is referenced by:  eupthi  28976  upgriseupth  28980  eupth0  28987  eupthres  28988  eupthp1  28989
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