Theorem cyclispthon

Description: A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017) (Revised by AV, 31-Jan-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Assertion	cyclispthon	⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ 0 ) ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		cyclispth	⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 ( Paths ‘ 𝐺 ) 𝑃 )
2		pthonpth	⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → 𝐹 ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) 𝑃 )
3	1 2	syl	⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) 𝑃 )
4		iscycl	⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
5	4	simprbi	⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) )
6	5	oveq2d	⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ 0 ) ) = ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
7	6	breqd	⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ( 𝐹 ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ 0 ) ) 𝑃 ↔ 𝐹 ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) 𝑃 ) )
8	3 7	mpbird	⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ 0 ) ) 𝑃 )