2spthd

Description: A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018) (Revised by AV, 24-Jan-2021) (Revised by AV, 24-Mar-2021) (Proof shortened by AV, 30-Oct-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	2spthd	⊢ ( 𝜑 → 𝐹 ( SPaths ‘ 𝐺 ) 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	2trld	⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )
11		3anan32	⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) ↔ ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ∧ 𝐴 ≠ 𝐶 ) )
12	4 9 11	sylanbrc	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) )
13		funcnvs3	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) ) → Fun ^◡ ⟨“ 𝐴 𝐵 𝐶 ”⟩ )
14	3 12 13	syl2anc	⊢ ( 𝜑 → Fun ^◡ ⟨“ 𝐴 𝐵 𝐶 ”⟩ )
15	1	a1i	⊢ ( 𝜑 → 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩ )
16	15	cnveqd	⊢ ( 𝜑 → ^◡ 𝑃 = ^◡ ⟨“ 𝐴 𝐵 𝐶 ”⟩ )
17	16	funeqd	⊢ ( 𝜑 → ( Fun ^◡ 𝑃 ↔ Fun ^◡ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ) )
18	14 17	mpbird	⊢ ( 𝜑 → Fun ^◡ 𝑃 )
19		isspth	⊢ ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ^◡ 𝑃 ) )
20	10 18 19	sylanbrc	⊢ ( 𝜑 → 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 )

Metamath Proof Explorer

Theorem 2spthd

Proof