Theorem numclwwlk2lem3

Description: In a friendship graph, the size of the set of walks of length N starting with a fixed vertex X and ending not at this vertex equals the size of the set of all closed walks of length ( N + 2 ) starting at this vertex X and not having this vertex as last but 2 vertex. (Contributed by Alexander van der Vekens, 6-Oct-2018) (Revised by AV, 31-May-2021) (Proof shortened by AV, 3-Nov-2022)

	Ref	Expression
Hypotheses	numclwwlk.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	numclwwlk.q	⊢ 𝑄 = ( 𝑣 ∈ 𝑉 , 𝑛 ∈ ℕ ↦ { 𝑤 ∈ ( 𝑛 WWalksN 𝐺 ) ∣ ( ( 𝑤 ‘ 0 ) = 𝑣 ∧ ( lastS ‘ 𝑤 ) ≠ 𝑣 ) } )
	numclwwlk.h	⊢ 𝐻 = ( 𝑣 ∈ 𝑉 , 𝑛 ∈ ( ℤ≥ ‘ 2 ) ↦ { 𝑤 ∈ ( 𝑣 ( ClWWalksNOn ‘ 𝐺 ) 𝑛 ) ∣ ( 𝑤 ‘ ( 𝑛 − 2 ) ) ≠ 𝑣 } )
Assertion	numclwwlk2lem3	⊢ ( ( 𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ♯ ‘ ( 𝑋 𝑄 𝑁 ) ) = ( ♯ ‘ ( 𝑋 𝐻 ( 𝑁 + 2 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		numclwwlk.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		numclwwlk.q	⊢ 𝑄 = ( 𝑣 ∈ 𝑉 , 𝑛 ∈ ℕ ↦ { 𝑤 ∈ ( 𝑛 WWalksN 𝐺 ) ∣ ( ( 𝑤 ‘ 0 ) = 𝑣 ∧ ( lastS ‘ 𝑤 ) ≠ 𝑣 ) } )
3		numclwwlk.h	⊢ 𝐻 = ( 𝑣 ∈ 𝑉 , 𝑛 ∈ ( ℤ≥ ‘ 2 ) ↦ { 𝑤 ∈ ( 𝑣 ( ClWWalksNOn ‘ 𝐺 ) 𝑛 ) ∣ ( 𝑤 ‘ ( 𝑛 − 2 ) ) ≠ 𝑣 } )
4		ovexd	⊢ ( ( 𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑋 𝐻 ( 𝑁 + 2 ) ) ∈ V )
5		eqid	⊢ ( ℎ ∈ ( 𝑋 𝐻 ( 𝑁 + 2 ) ) ↦ ( ℎ prefix ( 𝑁 + 1 ) ) ) = ( ℎ ∈ ( 𝑋 𝐻 ( 𝑁 + 2 ) ) ↦ ( ℎ prefix ( 𝑁 + 1 ) ) )
6	1 2 3 5	numclwlk2lem2f1o	⊢ ( ( 𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ℎ ∈ ( 𝑋 𝐻 ( 𝑁 + 2 ) ) ↦ ( ℎ prefix ( 𝑁 + 1 ) ) ) : ( 𝑋 𝐻 ( 𝑁 + 2 ) ) –1-1-onto→ ( 𝑋 𝑄 𝑁 ) )
7	4 6	hasheqf1od	⊢ ( ( 𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ♯ ‘ ( 𝑋 𝐻 ( 𝑁 + 2 ) ) ) = ( ♯ ‘ ( 𝑋 𝑄 𝑁 ) ) )
8	7	eqcomd	⊢ ( ( 𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ♯ ‘ ( 𝑋 𝑄 𝑁 ) ) = ( ♯ ‘ ( 𝑋 𝐻 ( 𝑁 + 2 ) ) ) )