Theorem upgrwlkcompim

Description: Implications for the properties of the components of a walk in a pseudograph. (Contributed by Alexander van der Vekens, 23-Jun-2018) (Revised by AV, 14-Apr-2021)

	Ref	Expression
Hypotheses	upgrwlkcompim.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	upgrwlkcompim.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	upgrwlkcompim.1	⊢ 𝐹 = ( 1st ‘ 𝑊 )
	upgrwlkcompim.2	⊢ 𝑃 = ( 2nd ‘ 𝑊 )
Assertion	upgrwlkcompim	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝑊 ∈ ( Walks ‘ 𝐺 ) ) → ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) )

Proof

Step	Hyp	Ref	Expression
1		upgrwlkcompim.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		upgrwlkcompim.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3		upgrwlkcompim.1	⊢ 𝐹 = ( 1st ‘ 𝑊 )
4		upgrwlkcompim.2	⊢ 𝑃 = ( 2nd ‘ 𝑊 )
5		wlkcpr	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) ↔ ( 1st ‘ 𝑊 ) ( Walks ‘ 𝐺 ) ( 2nd ‘ 𝑊 ) )
6	3 4	breq12i	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 1st ‘ 𝑊 ) ( Walks ‘ 𝐺 ) ( 2nd ‘ 𝑊 ) )
7	5 6	bitr4i	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) ↔ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
8	1 2	upgriswlk	⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) )
9	8	biimpd	⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) )
10	7 9	biimtrid	⊢ ( 𝐺 ∈ UPGraph → ( 𝑊 ∈ ( Walks ‘ 𝐺 ) → ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) )
11	10	imp	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝑊 ∈ ( Walks ‘ 𝐺 ) ) → ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) )