Theorem wwlksnexthasheq

Description: The number of the extensions of a walk (as word) by an edge equals the number of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 23-Aug-2018) (Revised by AV, 19-Apr-2021) (Revised by AV, 27-Oct-2022)

	Ref	Expression
Hypotheses	wwlksnexthasheq.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	wwlksnexthasheq.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	wwlksnexthasheq	⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( ♯ ‘ { 𝑤 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ∣ ( ( 𝑤 prefix ( 𝑁 + 1 ) ) = 𝑊 ∧ { ( lastS ‘ 𝑊 ) , ( lastS ‘ 𝑤 ) } ∈ 𝐸 ) } ) = ( ♯ ‘ { 𝑛 ∈ 𝑉 ∣ { ( lastS ‘ 𝑊 ) , 𝑛 } ∈ 𝐸 } ) )

Proof

Step	Hyp	Ref	Expression
1		wwlksnexthasheq.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		wwlksnexthasheq.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3		ovex	⊢ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ∈ V
4	3	rabex	⊢ { 𝑤 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ∣ ( ( 𝑤 prefix ( 𝑁 + 1 ) ) = 𝑊 ∧ { ( lastS ‘ 𝑊 ) , ( lastS ‘ 𝑤 ) } ∈ 𝐸 ) } ∈ V
5	1 2	wwlksnextbij	⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ∃ 𝑓 𝑓 : { 𝑤 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ∣ ( ( 𝑤 prefix ( 𝑁 + 1 ) ) = 𝑊 ∧ { ( lastS ‘ 𝑊 ) , ( lastS ‘ 𝑤 ) } ∈ 𝐸 ) } –1-1-onto→ { 𝑛 ∈ 𝑉 ∣ { ( lastS ‘ 𝑊 ) , 𝑛 } ∈ 𝐸 } )
6		hasheqf1oi	⊢ ( { 𝑤 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ∣ ( ( 𝑤 prefix ( 𝑁 + 1 ) ) = 𝑊 ∧ { ( lastS ‘ 𝑊 ) , ( lastS ‘ 𝑤 ) } ∈ 𝐸 ) } ∈ V → ( ∃ 𝑓 𝑓 : { 𝑤 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ∣ ( ( 𝑤 prefix ( 𝑁 + 1 ) ) = 𝑊 ∧ { ( lastS ‘ 𝑊 ) , ( lastS ‘ 𝑤 ) } ∈ 𝐸 ) } –1-1-onto→ { 𝑛 ∈ 𝑉 ∣ { ( lastS ‘ 𝑊 ) , 𝑛 } ∈ 𝐸 } → ( ♯ ‘ { 𝑤 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ∣ ( ( 𝑤 prefix ( 𝑁 + 1 ) ) = 𝑊 ∧ { ( lastS ‘ 𝑊 ) , ( lastS ‘ 𝑤 ) } ∈ 𝐸 ) } ) = ( ♯ ‘ { 𝑛 ∈ 𝑉 ∣ { ( lastS ‘ 𝑊 ) , 𝑛 } ∈ 𝐸 } ) ) )
7	4 5 6	mpsyl	⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( ♯ ‘ { 𝑤 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ∣ ( ( 𝑤 prefix ( 𝑁 + 1 ) ) = 𝑊 ∧ { ( lastS ‘ 𝑊 ) , ( lastS ‘ 𝑤 ) } ∈ 𝐸 ) } ) = ( ♯ ‘ { 𝑛 ∈ 𝑉 ∣ { ( lastS ‘ 𝑊 ) , 𝑛 } ∈ 𝐸 } ) )