Theorem numclwwlk1lem2f1o

Description: T is a 1-1 onto function. (Contributed by Alexander van der Vekens, 26-Sep-2018) (Revised by AV, 29-May-2021) (Revised by AV, 6-Mar-2022)

	Ref	Expression
Hypotheses	extwwlkfab.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	extwwlkfab.c	⊢ 𝐶 = ( 𝑣 ∈ 𝑉 , 𝑛 ∈ ( ℤ≥ ‘ 2 ) ↦ { 𝑤 ∈ ( 𝑣 ( ClWWalksNOn ‘ 𝐺 ) 𝑛 ) ∣ ( 𝑤 ‘ ( 𝑛 − 2 ) ) = 𝑣 } )
	extwwlkfab.f	⊢ 𝐹 = ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) ( 𝑁 − 2 ) )
	numclwwlk.t	⊢ 𝑇 = ( 𝑢 ∈ ( 𝑋 𝐶 𝑁 ) ↦ ⟨ ( 𝑢 prefix ( 𝑁 − 2 ) ) , ( 𝑢 ‘ ( 𝑁 − 1 ) ) ⟩ )
Assertion	numclwwlk1lem2f1o	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ( ℤ≥ ‘ 3 ) ) → 𝑇 : ( 𝑋 𝐶 𝑁 ) –1-1-onto→ ( 𝐹 × ( 𝐺 NeighbVtx 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		extwwlkfab.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		extwwlkfab.c	⊢ 𝐶 = ( 𝑣 ∈ 𝑉 , 𝑛 ∈ ( ℤ≥ ‘ 2 ) ↦ { 𝑤 ∈ ( 𝑣 ( ClWWalksNOn ‘ 𝐺 ) 𝑛 ) ∣ ( 𝑤 ‘ ( 𝑛 − 2 ) ) = 𝑣 } )
3		extwwlkfab.f	⊢ 𝐹 = ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) ( 𝑁 − 2 ) )
4		numclwwlk.t	⊢ 𝑇 = ( 𝑢 ∈ ( 𝑋 𝐶 𝑁 ) ↦ ⟨ ( 𝑢 prefix ( 𝑁 − 2 ) ) , ( 𝑢 ‘ ( 𝑁 − 1 ) ) ⟩ )
5	1 2 3 4	numclwwlk1lem2f1	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ( ℤ≥ ‘ 3 ) ) → 𝑇 : ( 𝑋 𝐶 𝑁 ) –1-1→ ( 𝐹 × ( 𝐺 NeighbVtx 𝑋 ) ) )
6	1 2 3 4	numclwwlk1lem2fo	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ( ℤ≥ ‘ 3 ) ) → 𝑇 : ( 𝑋 𝐶 𝑁 ) –onto→ ( 𝐹 × ( 𝐺 NeighbVtx 𝑋 ) ) )
7		df-f1o	⊢ ( 𝑇 : ( 𝑋 𝐶 𝑁 ) –1-1-onto→ ( 𝐹 × ( 𝐺 NeighbVtx 𝑋 ) ) ↔ ( 𝑇 : ( 𝑋 𝐶 𝑁 ) –1-1→ ( 𝐹 × ( 𝐺 NeighbVtx 𝑋 ) ) ∧ 𝑇 : ( 𝑋 𝐶 𝑁 ) –onto→ ( 𝐹 × ( 𝐺 NeighbVtx 𝑋 ) ) ) )
8	5 6 7	sylanbrc	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ( ℤ≥ ‘ 3 ) ) → 𝑇 : ( 𝑋 𝐶 𝑁 ) –1-1-onto→ ( 𝐹 × ( 𝐺 NeighbVtx 𝑋 ) ) )