Theorem clwwlken

Description: The set of closed walks of a fixed length represented by walks (as words) and the set of closed walks (as words) of the fixed length are equinumerous. (Contributed by AV, 5-Jun-2022) (Proof shortened by AV, 2-Nov-2022)

	Ref	Expression
Assertion	clwwlken	⊢ ( 𝑁 ∈ ℕ → { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } ≈ ( 𝑁 ClWWalksN 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		ovex	⊢ ( 𝑁 WWalksN 𝐺 ) ∈ V
2	1	rabex	⊢ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } ∈ V
3		ovex	⊢ ( 𝑁 ClWWalksN 𝐺 ) ∈ V
4		eqid	⊢ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } = { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) }
5		eqid	⊢ ( 𝑐 ∈ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } ↦ ( 𝑐 prefix 𝑁 ) ) = ( 𝑐 ∈ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } ↦ ( 𝑐 prefix 𝑁 ) )
6	4 5	clwwlkf1o	⊢ ( 𝑁 ∈ ℕ → ( 𝑐 ∈ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } ↦ ( 𝑐 prefix 𝑁 ) ) : { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } –1-1-onto→ ( 𝑁 ClWWalksN 𝐺 ) )
7		f1oen2g	⊢ ( ( { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } ∈ V ∧ ( 𝑁 ClWWalksN 𝐺 ) ∈ V ∧ ( 𝑐 ∈ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } ↦ ( 𝑐 prefix 𝑁 ) ) : { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } –1-1-onto→ ( 𝑁 ClWWalksN 𝐺 ) ) → { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } ≈ ( 𝑁 ClWWalksN 𝐺 ) )
8	2 3 6 7	mp3an12i	⊢ ( 𝑁 ∈ ℕ → { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } ≈ ( 𝑁 ClWWalksN 𝐺 ) )