Theorem 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 ) ) } ) )