Theorem upgrf1istrl

Description: Properties of a pair of a one-to-one function into the set of indices of edges and a function into the set of vertices to be a trail in a pseudograph. (Contributed by Alexander van der Vekens, 20-Oct-2017) (Revised by AV, 7-Jan-2021) (Revised by AV, 29-Oct-2021)

	Ref	Expression
Hypotheses	upgrtrls.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	upgrtrls.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion	upgrf1istrl	⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) )

Proof

Step	Hyp	Ref	Expression
1		upgrtrls.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		upgrtrls.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3	1 2	upgristrl	⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( ( 𝐹 ∈ Word dom 𝐼 ∧ Fun ◡ 𝐹 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) )
4		iswrdb	⊢ ( 𝐹 ∈ Word dom 𝐼 ↔ 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐼 )
5	4	a1i	⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ∈ Word dom 𝐼 ↔ 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐼 ) )
6	5	anbi1d	⊢ ( 𝐺 ∈ UPGraph → ( ( 𝐹 ∈ Word dom 𝐼 ∧ Fun ◡ 𝐹 ) ↔ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐼 ∧ Fun ◡ 𝐹 ) ) )
7		df-f1	⊢ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 ↔ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐼 ∧ Fun ◡ 𝐹 ) )
8	6 7	bitr4di	⊢ ( 𝐺 ∈ UPGraph → ( ( 𝐹 ∈ Word dom 𝐼 ∧ Fun ◡ 𝐹 ) ↔ 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 ) )
9	8	3anbi1d	⊢ ( 𝐺 ∈ UPGraph → ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ Fun ◡ 𝐹 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ↔ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) )
10	3 9	bitrd	⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) )