Theorem 0spth

Description: A pair of an empty set (of edges) and a second set (of vertices) is a simple path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017) (Revised by AV, 18-Jan-2021) (Revised by AV, 30-Oct-2021)

	Ref	Expression
Hypothesis	0pth.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	0spth	⊢ ( 𝐺 ∈ 𝑊 → ( ∅ ( SPaths ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		0pth.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	0trl	⊢ ( 𝐺 ∈ 𝑊 → ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )
3	2	anbi1d	⊢ ( 𝐺 ∈ 𝑊 → ( ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝑃 ) ↔ ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ Fun ◡ 𝑃 ) ) )
4		isspth	⊢ ( ∅ ( SPaths ‘ 𝐺 ) 𝑃 ↔ ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝑃 ) )
5		fz0sn	⊢ ( 0 ... 0 ) = { 0 }
6	5	feq2i	⊢ ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ↔ 𝑃 : { 0 } ⟶ 𝑉 )
7		c0ex	⊢ 0 ∈ V
8	7	fsn2	⊢ ( 𝑃 : { 0 } ⟶ 𝑉 ↔ ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ∧ 𝑃 = { ⟨ 0 , ( 𝑃 ‘ 0 ) ⟩ } ) )
9		funcnvsn	⊢ Fun ◡ { ⟨ 0 , ( 𝑃 ‘ 0 ) ⟩ }
10		cnveq	⊢ ( 𝑃 = { ⟨ 0 , ( 𝑃 ‘ 0 ) ⟩ } → ◡ 𝑃 = ◡ { ⟨ 0 , ( 𝑃 ‘ 0 ) ⟩ } )
11	10	funeqd	⊢ ( 𝑃 = { ⟨ 0 , ( 𝑃 ‘ 0 ) ⟩ } → ( Fun ◡ 𝑃 ↔ Fun ◡ { ⟨ 0 , ( 𝑃 ‘ 0 ) ⟩ } ) )
12	9 11	mpbiri	⊢ ( 𝑃 = { ⟨ 0 , ( 𝑃 ‘ 0 ) ⟩ } → Fun ◡ 𝑃 )
13	8 12	simplbiim	⊢ ( 𝑃 : { 0 } ⟶ 𝑉 → Fun ◡ 𝑃 )
14	6 13	sylbi	⊢ ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 → Fun ◡ 𝑃 )
15	14	pm4.71i	⊢ ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ↔ ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ Fun ◡ 𝑃 ) )
16	3 4 15	3bitr4g	⊢ ( 𝐺 ∈ 𝑊 → ( ∅ ( SPaths ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )