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Metamath Proof Explorer

Theorem uspgrf

Description: The edge function of a simple pseudograph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017) (Revised by AV, 13-Oct-2020)

Ref Expression
Hypotheses isuspgr.v ⊢ V = Vtx ⁡ G
isuspgr.e ⊢ E = iEdg ⁡ G
Assertion uspgrf ⊢ G ∈ USHGraph → E : dom ⁡ E ⟶ 1-1 x ∈ 𝒫 V ∖ ∅ | x ≤ 2

Proof

Step Hyp Ref Expression
1 isuspgr.v ⊢ V = Vtx ⁡ G
2 isuspgr.e ⊢ E = iEdg ⁡ G
3 1 2 isuspgr ⊢ G ∈ USHGraph → G ∈ USHGraph ↔ E : dom ⁡ E ⟶ 1-1 x ∈ 𝒫 V ∖ ∅ | x ≤ 2
4 3 ibi ⊢ G ∈ USHGraph → E : dom ⁡ E ⟶ 1-1 x ∈ 𝒫 V ∖ ∅ | x ≤ 2