Theorem 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 ) )