Theorem wwlksn

Description: The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018) (Revised by AV, 8-Apr-2021)

	Ref	Expression
Assertion	wwlksn	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 WWalksN 𝐺 ) = { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝑔 = 𝐺 → ( WWalks ‘ 𝑔 ) = ( WWalks ‘ 𝐺 ) )
2	1	adantl	⊢ ( ( 𝑛 = 𝑁 ∧ 𝑔 = 𝐺 ) → ( WWalks ‘ 𝑔 ) = ( WWalks ‘ 𝐺 ) )
3		oveq1	⊢ ( 𝑛 = 𝑁 → ( 𝑛 + 1 ) = ( 𝑁 + 1 ) )
4	3	eqeq2d	⊢ ( 𝑛 = 𝑁 → ( ( ♯ ‘ 𝑤 ) = ( 𝑛 + 1 ) ↔ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) )
5	4	adantr	⊢ ( ( 𝑛 = 𝑁 ∧ 𝑔 = 𝐺 ) → ( ( ♯ ‘ 𝑤 ) = ( 𝑛 + 1 ) ↔ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) )
6	2 5	rabeqbidv	⊢ ( ( 𝑛 = 𝑁 ∧ 𝑔 = 𝐺 ) → { 𝑤 ∈ ( WWalks ‘ 𝑔 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑛 + 1 ) } = { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } )
7		df-wwlksn	⊢ WWalksN = ( 𝑛 ∈ ℕ0 , 𝑔 ∈ V ↦ { 𝑤 ∈ ( WWalks ‘ 𝑔 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑛 + 1 ) } )
8		fvex	⊢ ( WWalks ‘ 𝐺 ) ∈ V
9	8	rabex	⊢ { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } ∈ V
10	6 7 9	ovmpoa	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) → ( 𝑁 WWalksN 𝐺 ) = { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } )
11	10	expcom	⊢ ( 𝐺 ∈ V → ( 𝑁 ∈ ℕ0 → ( 𝑁 WWalksN 𝐺 ) = { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } ) )
12	7	reldmmpo	⊢ Rel dom WWalksN
13	12	ovprc2	⊢ ( ¬ 𝐺 ∈ V → ( 𝑁 WWalksN 𝐺 ) = ∅ )
14		fvprc	⊢ ( ¬ 𝐺 ∈ V → ( WWalks ‘ 𝐺 ) = ∅ )
15	14	rabeqdv	⊢ ( ¬ 𝐺 ∈ V → { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } = { 𝑤 ∈ ∅ ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } )
16		rab0	⊢ { 𝑤 ∈ ∅ ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } = ∅
17	15 16	eqtrdi	⊢ ( ¬ 𝐺 ∈ V → { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } = ∅ )
18	13 17	eqtr4d	⊢ ( ¬ 𝐺 ∈ V → ( 𝑁 WWalksN 𝐺 ) = { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } )
19	18	a1d	⊢ ( ¬ 𝐺 ∈ V → ( 𝑁 ∈ ℕ0 → ( 𝑁 WWalksN 𝐺 ) = { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } ) )
20	11 19	pm2.61i	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 WWalksN 𝐺 ) = { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } )