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Metamath Proof Explorer

Theorem usgrfun

Description: The edge function of a simple graph is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017) (Revised by AV, 13-Oct-2020)

Ref Expression
Assertion usgrfun ⊢ G ∈ USGraph → Fun ⁡ iEdg ⁡ G

Proof

Step Hyp Ref Expression
1 eqid ⊢ Vtx ⁡ G = Vtx ⁡ G
2 eqid ⊢ iEdg ⁡ G = iEdg ⁡ G
3 1 2 usgrfs ⊢ G ∈ USGraph → iEdg ⁡ G : dom ⁡ iEdg ⁡ G ⟶ 1-1 x ∈ 𝒫 Vtx ⁡ G | x = 2
4 f1fun ⊢ iEdg ⁡ G : dom ⁡ iEdg ⁡ G ⟶ 1-1 x ∈ 𝒫 Vtx ⁡ G | x = 2 → Fun ⁡ iEdg ⁡ G
5 3 4 syl ⊢ G ∈ USGraph → Fun ⁡ iEdg ⁡ G