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Metamath Proof Explorer

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 ) ) } ) ) )