Theorem 0trlon

Description: A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017) (Revised by AV, 8-Jan-2021) (Revised by AV, 23-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Hypothesis	0wlk.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	0trlon	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ∅ ( 𝑁 ( TrailsOn ‘ 𝐺 ) 𝑁 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		0wlk.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	0wlkon	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ∅ ( 𝑁 ( WalksOn ‘ 𝐺 ) 𝑁 ) 𝑃 )
3		simpl	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 )
4	1	0wlkonlem1	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ( 𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ) )
5	1	1vgrex	⊢ ( 𝑁 ∈ 𝑉 → 𝐺 ∈ V )
6	5	adantr	⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ) → 𝐺 ∈ V )
7	1	0trl	⊢ ( 𝐺 ∈ V → ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )
8	4 6 7	3syl	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )
9	3 8	mpbird	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ∅ ( Trails ‘ 𝐺 ) 𝑃 )
10		0ex	⊢ ∅ ∈ V
11	10	a1i	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ∅ ∈ V )
12	1	0wlkonlem2	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → 𝑃 ∈ ( 𝑉 ↑pm ( 0 ... 0 ) ) )
13	1	istrlson	⊢ ( ( ( 𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ) ∧ ( ∅ ∈ V ∧ 𝑃 ∈ ( 𝑉 ↑pm ( 0 ... 0 ) ) ) ) → ( ∅ ( 𝑁 ( TrailsOn ‘ 𝐺 ) 𝑁 ) 𝑃 ↔ ( ∅ ( 𝑁 ( WalksOn ‘ 𝐺 ) 𝑁 ) 𝑃 ∧ ∅ ( Trails ‘ 𝐺 ) 𝑃 ) ) )
14	4 11 12 13	syl12anc	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ( ∅ ( 𝑁 ( TrailsOn ‘ 𝐺 ) 𝑁 ) 𝑃 ↔ ( ∅ ( 𝑁 ( WalksOn ‘ 𝐺 ) 𝑁 ) 𝑃 ∧ ∅ ( Trails ‘ 𝐺 ) 𝑃 ) ) )
15	2 9 14	mpbir2and	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ∅ ( 𝑁 ( TrailsOn ‘ 𝐺 ) 𝑁 ) 𝑃 )