Theorem 2pthond

Description: A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017) (Revised by AV, 24-Jan-2021) (Proof shortened by AV, 30-Jan-2021) (Revised by AV, 24-Mar-2021)

	Ref	Expression
Hypotheses	2wlkd.p	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
	2wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
	2wlkd.s	⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
	2wlkd.n	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
	2wlkd.e	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
	2wlkd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	2wlkd.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	2trld.n	⊢ ( 𝜑 → 𝐽 ≠ 𝐾 )
	2spthd.n	⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )
Assertion	2pthond	⊢ ( 𝜑 → 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐶 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		2wlkd.p	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
2		2wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
3		2wlkd.s	⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
4		2wlkd.n	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
5		2wlkd.e	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
6		2wlkd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
7		2wlkd.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
8		2trld.n	⊢ ( 𝜑 → 𝐽 ≠ 𝐾 )
9		2spthd.n	⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )
10	1 2 3 4 5 6 7 8	2trlond	⊢ ( 𝜑 → 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐶 ) 𝑃 )
11	1 2 3 4 5 6 7 8 9	2spthd	⊢ ( 𝜑 → 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 )
12		3simpb	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
13	3 12	syl	⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
14		s2cli	⊢ ⟨“ 𝐽 𝐾 ”⟩ ∈ Word V
15	2 14	eqeltri	⊢ 𝐹 ∈ Word V
16		s3cli	⊢ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ Word V
17	1 16	eqeltri	⊢ 𝑃 ∈ Word V
18	15 17	pm3.2i	⊢ ( 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V )
19	6	isspthson	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V ) ) → ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐶 ) 𝑃 ↔ ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐶 ) 𝑃 ∧ 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ) ) )
20	13 18 19	sylancl	⊢ ( 𝜑 → ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐶 ) 𝑃 ↔ ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐶 ) 𝑃 ∧ 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ) ) )
21	10 11 20	mpbir2and	⊢ ( 𝜑 → 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐶 ) 𝑃 )