Theorem 0cycl

Description: A pair of an empty set (of edges) and a second set (of vertices) is a cycle if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017) (Revised by AV, 31-Jan-2021) (Revised by AV, 30-Oct-2021)

	Ref	Expression
Assertion	0cycl	⊢ ( 𝐺 ∈ 𝑊 → ( ∅ ( Cycles ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ ( Vtx ‘ 𝐺 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2	1	0pth	⊢ ( 𝐺 ∈ 𝑊 → ( ∅ ( Paths ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ ( Vtx ‘ 𝐺 ) ) )
3	2	anbi1d	⊢ ( 𝐺 ∈ 𝑊 → ( ( ∅ ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ ∅ ) ) ) ↔ ( 𝑃 : ( 0 ... 0 ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ ∅ ) ) ) ) )
4		iscycl	⊢ ( ∅ ( Cycles ‘ 𝐺 ) 𝑃 ↔ ( ∅ ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ ∅ ) ) ) )
5		hash0	⊢ ( ♯ ‘ ∅ ) = 0
6	5	eqcomi	⊢ 0 = ( ♯ ‘ ∅ )
7	6	fveq2i	⊢ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ ∅ ) )
8	7	biantru	⊢ ( 𝑃 : ( 0 ... 0 ) ⟶ ( Vtx ‘ 𝐺 ) ↔ ( 𝑃 : ( 0 ... 0 ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ ∅ ) ) ) )
9	3 4 8	3bitr4g	⊢ ( 𝐺 ∈ 𝑊 → ( ∅ ( Cycles ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ ( Vtx ‘ 𝐺 ) ) )