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Database GRAPH THEORY Eulerian paths and the Konigsberg Bridge problem Eulerian paths upgreupthi
Description: Properties of an Eulerian path in a pseudograph. (Contributed by Mario
Carneiro , 12-Mar-2015) (Revised by AV , 18-Feb-2021) (Proof shortened by AV , 30-Oct-2021)
Ref
Expression
Hypotheses
eupths.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
upgriseupth.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion
upgreupthi ⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 ) → ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1 -onto → dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) )
Proof
Step
Hyp
Ref
Expression
1
eupths.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2
upgriseupth.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
3
1 2
upgriseupth ⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1 -onto → dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) )
4
3
biimpa ⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 ) → ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1 -onto → dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) )