Metamath Proof Explorer

Theorem uspgrupgr

Description: A simple pseudograph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017) (Revised by AV, 15-Oct-2020)

Ref Expression
Assertion uspgrupgr ⊢ G ∈ USHGraph → G ∈ UPGraph

Proof

Step Hyp Ref Expression
1 eqid ⊢ Vtx ⁡ G = Vtx ⁡ G
2 eqid ⊢ iEdg ⁡ G = iEdg ⁡ G
3 1 2 isuspgr ⊢ G ∈ USHGraph → G ∈ USHGraph ↔ iEdg ⁡ G : dom ⁡ iEdg ⁡ G ⟶ 1-1 x ∈ 𝒫 Vtx ⁡ G ∖ ∅ | x ≤ 2
4 f1f ⊢ iEdg ⁡ G : dom ⁡ iEdg ⁡ G ⟶ 1-1 x ∈ 𝒫 Vtx ⁡ G ∖ ∅ | x ≤ 2 → iEdg ⁡ G : dom ⁡ iEdg ⁡ G ⟶ x ∈ 𝒫 Vtx ⁡ G ∖ ∅ | x ≤ 2
5 3 4 biimtrdi ⊢ G ∈ USHGraph → G ∈ USHGraph → iEdg ⁡ G : dom ⁡ iEdg ⁡ G ⟶ x ∈ 𝒫 Vtx ⁡ G ∖ ∅ | x ≤ 2
6 1 2 isupgr ⊢ G ∈ USHGraph → G ∈ UPGraph ↔ iEdg ⁡ G : dom ⁡ iEdg ⁡ G ⟶ x ∈ 𝒫 Vtx ⁡ G ∖ ∅ | x ≤ 2
7 5 6 sylibrd ⊢ G ∈ USHGraph → G ∈ USHGraph → G ∈ UPGraph
8 7 pm2.43i ⊢ G ∈ USHGraph → G ∈ UPGraph