Theorem clwwlknon2

Description: The set of closed walks on vertex X of length 2 in a graph G as words over the set of vertices. (Contributed by AV, 5-Mar-2022) (Revised by AV, 25-Mar-2022)

	Ref	Expression
Hypothesis	clwwlknon2.c	⊢ 𝐶 = ( ClWWalksNOn ‘ 𝐺 )
Assertion	clwwlknon2	⊢ ( 𝑋 𝐶 2 ) = { 𝑤 ∈ ( 2 ClWWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑋 }

Proof

Step	Hyp	Ref	Expression
1		clwwlknon2.c	⊢ 𝐶 = ( ClWWalksNOn ‘ 𝐺 )
2	1	oveqi	⊢ ( 𝑋 𝐶 2 ) = ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 2 )
3		clwwlknon	⊢ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 2 ) = { 𝑤 ∈ ( 2 ClWWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑋 }
4	2 3	eqtri	⊢ ( 𝑋 𝐶 2 ) = { 𝑤 ∈ ( 2 ClWWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑋 }