Theorem upgrtrls

Description: The set of trails in a pseudograph, definition of walks expanded. (Contributed by Alexander van der Vekens, 20-Oct-2017) (Revised by AV, 7-Jan-2021)

	Ref	Expression
Hypotheses	upgrtrls.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	upgrtrls.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion	upgrtrls	⊢ ( 𝐺 ∈ UPGraph → ( Trails ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( ( 𝑓 ∈ Word dom 𝐼 ∧ Fun ◡ 𝑓 ) ∧ 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		trlsfval	⊢ ( Trails ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ Fun ◡ 𝑓 ) }
4	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 ) ) } ) ) )
5	4	anbi1d	⊢ ( 𝐺 ∈ UPGraph → ( ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ Fun ◡ 𝑓 ) ↔ ( ( 𝑓 ∈ Word dom 𝐼 ∧ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) ∧ Fun ◡ 𝑓 ) ) )
6		an32	⊢ ( ( ( 𝑓 ∈ Word dom 𝐼 ∧ ( 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) ) ∧ Fun ◡ 𝑓 ) ↔ ( ( 𝑓 ∈ Word dom 𝐼 ∧ Fun ◡ 𝑓 ) ∧ ( 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) ) )
7		3anass	⊢ ( ( 𝑓 ∈ Word dom 𝐼 ∧ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) ↔ ( 𝑓 ∈ Word dom 𝐼 ∧ ( 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) ) )
8	7	anbi1i	⊢ ( ( ( 𝑓 ∈ Word dom 𝐼 ∧ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) ∧ Fun ◡ 𝑓 ) ↔ ( ( 𝑓 ∈ Word dom 𝐼 ∧ ( 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) ) ∧ Fun ◡ 𝑓 ) )
9		3anass	⊢ ( ( ( 𝑓 ∈ Word dom 𝐼 ∧ Fun ◡ 𝑓 ) ∧ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) ↔ ( ( 𝑓 ∈ Word dom 𝐼 ∧ Fun ◡ 𝑓 ) ∧ ( 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) ) )
10	6 8 9	3bitr4i	⊢ ( ( ( 𝑓 ∈ Word dom 𝐼 ∧ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) ∧ Fun ◡ 𝑓 ) ↔ ( ( 𝑓 ∈ Word dom 𝐼 ∧ Fun ◡ 𝑓 ) ∧ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) )
11	5 10	bitrdi	⊢ ( 𝐺 ∈ UPGraph → ( ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ Fun ◡ 𝑓 ) ↔ ( ( 𝑓 ∈ Word dom 𝐼 ∧ Fun ◡ 𝑓 ) ∧ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) ) )
12	11	opabbidv	⊢ ( 𝐺 ∈ UPGraph → { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ Fun ◡ 𝑓 ) } = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( ( 𝑓 ∈ Word dom 𝐼 ∧ Fun ◡ 𝑓 ) ∧ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) } )
13	3 12	syl5eq	⊢ ( 𝐺 ∈ UPGraph → ( Trails ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( ( 𝑓 ∈ Word dom 𝐼 ∧ Fun ◡ 𝑓 ) ∧ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ) } )