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Metamath Proof Explorer

Theorem 3trld

Description: Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017) (Revised by AV, 8-Feb-2021) (Revised by AV, 24-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

Ref Expression
Hypotheses 3wlkd.p ⊢ P = ⟨“ ABCD ”⟩
3wlkd.f ⊢ F = ⟨“ JKL ”⟩
3wlkd.s ⊢ φ → A ∈ V ∧ B ∈ V ∧ C ∈ V ∧ D ∈ V
3wlkd.n ⊢ φ → A ≠ B ∧ A ≠ C ∧ B ≠ C ∧ B ≠ D ∧ C ≠ D
3wlkd.e ⊢ φ → A B ⊆ I ⁡ J ∧ B C ⊆ I ⁡ K ∧ C D ⊆ I ⁡ L
3wlkd.v ⊢ V = Vtx ⁡ G
3wlkd.i ⊢ I = iEdg ⁡ G
3trld.n ⊢ φ → J ≠ K ∧ J ≠ L ∧ K ≠ L
Assertion 3trld ⊢ φ → F Trails ⁡ G P

Proof

Step Hyp Ref Expression
1 3wlkd.p ⊢ P = ⟨“ ABCD ”⟩
2 3wlkd.f ⊢ F = ⟨“ JKL ”⟩
3 3wlkd.s ⊢ φ → A ∈ V ∧ B ∈ V ∧ C ∈ V ∧ D ∈ V
4 3wlkd.n ⊢ φ → A ≠ B ∧ A ≠ C ∧ B ≠ C ∧ B ≠ D ∧ C ≠ D
5 3wlkd.e ⊢ φ → A B ⊆ I ⁡ J ∧ B C ⊆ I ⁡ K ∧ C D ⊆ I ⁡ L
6 3wlkd.v ⊢ V = Vtx ⁡ G
7 3wlkd.i ⊢ I = iEdg ⁡ G
8 3trld.n ⊢ φ → J ≠ K ∧ J ≠ L ∧ K ≠ L
9 1 2 3 4 5 6 7 3wlkd ⊢ φ → F Walks ⁡ G P
10 1 2 3 4 5 3wlkdlem7 ⊢ φ → J ∈ V ∧ K ∈ V ∧ L ∈ V
11 funcnvs3 ⊢ J ∈ V ∧ K ∈ V ∧ L ∈ V ∧ J ≠ K ∧ J ≠ L ∧ K ≠ L → Fun ⁡ ⟨“ JKL ”⟩ -1
12 10 8 11 syl2anc ⊢ φ → Fun ⁡ ⟨“ JKL ”⟩ -1
13 2 cnveqi ⊢ F -1 = ⟨“ JKL ”⟩ -1
14 13 funeqi ⊢ Fun ⁡ F -1 ↔ Fun ⁡ ⟨“ JKL ”⟩ -1
15 12 14 sylibr ⊢ φ → Fun ⁡ F -1
16 istrl ⊢ F Trails ⁡ G P ↔ F Walks ⁡ G P ∧ Fun ⁡ F -1
17 9 15 16 sylanbrc ⊢ φ → F Trails ⁡ G P