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Metamath Proof Explorer

Theorem 2trld

Description: Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017) (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 )
Assertion 2trld ( 𝜑 → 𝐹 ( Trails ‘ 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 1 2 3 4 5 6 7 2wlkd ⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
10 1 2 3 4 5 2wlkdlem7 ⊢ ( 𝜑 → ( 𝐽 ∈ V ∧ 𝐾 ∈ V ) )
11 df-3an ⊢ ( ( 𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐽 ≠ 𝐾 ) ↔ ( ( 𝐽 ∈ V ∧ 𝐾 ∈ V ) ∧ 𝐽 ≠ 𝐾 ) )
12 10 8 11 sylanbrc ⊢ ( 𝜑 → ( 𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐽 ≠ 𝐾 ) )
13 funcnvs2 ⊢ ( ( 𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐽 ≠ 𝐾 ) → Fun ◡ ⟨“ 𝐽 𝐾 ”⟩ )
14 12 13 syl ⊢ ( 𝜑 → Fun ◡ ⟨“ 𝐽 𝐾 ”⟩ )
15 2 cnveqi ⊢ ◡ 𝐹 = ◡ ⟨“ 𝐽 𝐾 ”⟩
16 15 funeqi ⊢ ( Fun ◡ 𝐹 ↔ Fun ◡ ⟨“ 𝐽 𝐾 ”⟩ )
17 14 16 sylibr ⊢ ( 𝜑 → Fun ◡ 𝐹 )
18 istrl ⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) )
19 9 17 18 sylanbrc ⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )