Theorem iswwlks

Description: A word over the set of vertices representing a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018) (Revised by AV, 8-Apr-2021)

	Ref	Expression
Hypotheses	wwlks.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	wwlks.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	iswwlks	⊢ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ↔ ( 𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		wwlks.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		wwlks.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3		neeq1	⊢ ( 𝑤 = 𝑊 → ( 𝑤 ≠ ∅ ↔ 𝑊 ≠ ∅ ) )
4		fveq2	⊢ ( 𝑤 = 𝑊 → ( ♯ ‘ 𝑤 ) = ( ♯ ‘ 𝑊 ) )
5	4	oveq1d	⊢ ( 𝑤 = 𝑊 → ( ( ♯ ‘ 𝑤 ) − 1 ) = ( ( ♯ ‘ 𝑊 ) − 1 ) )
6	5	oveq2d	⊢ ( 𝑤 = 𝑊 → ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) = ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) )
7		fveq1	⊢ ( 𝑤 = 𝑊 → ( 𝑤 ‘ 𝑖 ) = ( 𝑊 ‘ 𝑖 ) )
8		fveq1	⊢ ( 𝑤 = 𝑊 → ( 𝑤 ‘ ( 𝑖 + 1 ) ) = ( 𝑊 ‘ ( 𝑖 + 1 ) ) )
9	7 8	preq12d	⊢ ( 𝑤 = 𝑊 → { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } = { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } )
10	9	eleq1d	⊢ ( 𝑤 = 𝑊 → ( { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ↔ { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) )
11	6 10	raleqbidv	⊢ ( 𝑤 = 𝑊 → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ↔ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) )
12	3 11	anbi12d	⊢ ( 𝑤 = 𝑊 → ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) ↔ ( 𝑊 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) ) )
13	12	elrab	⊢ ( 𝑊 ∈ { 𝑤 ∈ Word 𝑉 ∣ ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) } ↔ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) ) )
14	1 2	wwlks	⊢ ( WWalks ‘ 𝐺 ) = { 𝑤 ∈ Word 𝑉 ∣ ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) }
15	14	eleq2i	⊢ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ↔ 𝑊 ∈ { 𝑤 ∈ Word 𝑉 ∣ ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) } )
16		3anan12	⊢ ( ( 𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) ↔ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) ) )
17	13 15 16	3bitr4i	⊢ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ↔ ( 𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) )