Theorem cycls

Description: The set of cycles (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017) (Revised by AV, 31-Jan-2021)

	Ref	Expression
Assertion	cycls	⊢ ( Cycles ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Paths ‘ 𝐺 ) 𝑝 ∧ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) }

Proof

Step	Hyp	Ref	Expression
1		biidd	⊢ ( ( ⊤ ∧ 𝑔 = 𝐺 ) → ( ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ↔ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) )
2		wksv	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Walks ‘ 𝐺 ) 𝑝 } ∈ V
3		pthiswlk	⊢ ( 𝑓 ( Paths ‘ 𝐺 ) 𝑝 → 𝑓 ( Walks ‘ 𝐺 ) 𝑝 )
4	3	ssopab2i	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Paths ‘ 𝐺 ) 𝑝 } ⊆ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Walks ‘ 𝐺 ) 𝑝 }
5	2 4	ssexi	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Paths ‘ 𝐺 ) 𝑝 } ∈ V
6	5	a1i	⊢ ( ⊤ → { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Paths ‘ 𝐺 ) 𝑝 } ∈ V )
7		df-cycls	⊢ Cycles = ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Paths ‘ 𝑔 ) 𝑝 ∧ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) } )
8	1 6 7	fvmptopab	⊢ ( ⊤ → ( Cycles ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Paths ‘ 𝐺 ) 𝑝 ∧ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) } )
9	8	mptru	⊢ ( Cycles ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Paths ‘ 𝐺 ) 𝑝 ∧ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) }