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Database GRAPH THEORY Eulerian paths and the Konigsberg Bridge problem Eulerian paths eupthi
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)
Ref
Expression
Hypothesis
eupths.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion
eupthi ⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1 -onto → dom 𝐼 ) )
Proof
Step
Hyp
Ref
Expression
1
eupths.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2
1
iseupthf1o ⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1 -onto → dom 𝐼 ) )
3
2
biimpi ⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1 -onto → dom 𝐼 ) )