Metamath Proof Explorer

Theorem usgredg2

Description: The value of the "edge function" of a simple graph is a set containing two elements (the vertices the corresponding edge is connecting). (Contributed by Alexander van der Vekens, 11-Aug-2017) (Revised by AV, 16-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

Ref Expression
Hypothesis usgredg2.e ⊢ E = iEdg ⁡ G
Assertion usgredg2 ⊢ G ∈ USGraph ∧ X ∈ dom ⁡ E → E ⁡ X = 2

Proof

Step Hyp Ref Expression
1 usgredg2.e ⊢ E = iEdg ⁡ G
2 usgrumgr ⊢ G ∈ USGraph → G ∈ UMGraph
3 eqid ⊢ Vtx ⁡ G = Vtx ⁡ G
4 3 1 umgredg2 ⊢ G ∈ UMGraph ∧ X ∈ dom ⁡ E → E ⁡ X = 2
5 2 4 sylan ⊢ G ∈ USGraph ∧ X ∈ dom ⁡ E → E ⁡ X = 2