Metamath Proof Explorer

Theorem spthonprop

Description: Properties of a simple path between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018) (Revised by AV, 16-Jan-2021)

Ref Expression
Hypothesis pthsonfval.v ⊢ V = Vtx ⁡ G
Assertion spthonprop ⊢ F A SPathsOn ⁡ G B P → G ∈ V ∧ A ∈ V ∧ B ∈ V ∧ F ∈ V ∧ P ∈ V ∧ F A TrailsOn ⁡ G B P ∧ F SPaths ⁡ G P

Proof

Step Hyp Ref Expression
1 pthsonfval.v ⊢ V = Vtx ⁡ G
2 1 isspthson ⊢ A ∈ V ∧ B ∈ V ∧ F ∈ V ∧ P ∈ V → F A SPathsOn ⁡ G B P ↔ F A TrailsOn ⁡ G B P ∧ F SPaths ⁡ G P
3 2 3adantl1 ⊢ G ∈ V ∧ A ∈ V ∧ B ∈ V ∧ F ∈ V ∧ P ∈ V → F A SPathsOn ⁡ G B P ↔ F A TrailsOn ⁡ G B P ∧ F SPaths ⁡ G P
4 df-spthson ⊢ SPathsOn = g ∈ V ⟼ a ∈ Vtx ⁡ g , b ∈ Vtx ⁡ g ⟼ f p | f a TrailsOn ⁡ g b p ∧ f SPaths ⁡ g p
5 spthiswlk ⊢ f SPaths ⁡ G p → f Walks ⁡ G p
6 5 adantl ⊢ G ∈ V ∧ A ∈ V ∧ B ∈ V ∧ f SPaths ⁡ G p → f Walks ⁡ G p
7 1 3 4 6 wksonproplem ⊢ F A SPathsOn ⁡ G B P → G ∈ V ∧ A ∈ V ∧ B ∈ V ∧ F ∈ V ∧ P ∈ V ∧ F A TrailsOn ⁡ G B P ∧ F SPaths ⁡ G P