Theorem iswlkg

Description: Generalization of iswlk : Conditions for two classes to represent a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018) (Revised by AV, 1-Jan-2021)

	Ref	Expression
Hypotheses	iswlkg.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	iswlkg.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion	iswlkg	⊢ ( 𝐺 ∈ 𝑊 → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		iswlkg.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		iswlkg.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3		wlkv	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) )
4		3simpc	⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) )
5	3 4	syl	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) )
6	5	a1i	⊢ ( 𝐺 ∈ 𝑊 → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) )
7		elex	⊢ ( 𝐹 ∈ Word dom 𝐼 → 𝐹 ∈ V )
8		ovex	⊢ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∈ V
9	1	fvexi	⊢ 𝑉 ∈ V
10	8 9	fpm	⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 → 𝑃 ∈ ( 𝑉 ↑pm ( 0 ... ( ♯ ‘ 𝐹 ) ) ) )
11	10	elexd	⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 → 𝑃 ∈ V )
12	7 11	anim12i	⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) → ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) )
13	12	3adant3	⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) )
14	13	a1i	⊢ ( 𝐺 ∈ 𝑊 → ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) )
15	1 2	iswlk	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )
16	15	3expib	⊢ ( 𝐺 ∈ 𝑊 → ( ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) )
17	6 14 16	pm5.21ndd	⊢ ( 𝐺 ∈ 𝑊 → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )