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Metamath Proof Explorer

Theorem isusgrs

Description: The property of being a simple graph, simplified version of isusgr . (Contributed by Alexander van der Vekens, 13-Aug-2017) (Revised by AV, 13-Oct-2020) (Proof shortened by AV, 24-Nov-2020)

Ref Expression
Hypotheses isuspgr.v ⊢ V = Vtx ⁡ G
isuspgr.e ⊢ E = iEdg ⁡ G
Assertion isusgrs ⊢ G ∈ U → G ∈ USGraph ↔ 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 isusgr ⊢ G ∈ U → G ∈ USGraph ↔ E : dom ⁡ E ⟶ 1-1 x ∈ 𝒫 V ∖ ∅ | x = 2
4 prprrab ⊢ x ∈ 𝒫 V ∖ ∅ | x = 2 = x ∈ 𝒫 V | x = 2
5 f1eq3 ⊢ x ∈ 𝒫 V ∖ ∅ | x = 2 = x ∈ 𝒫 V | x = 2 → E : dom ⁡ E ⟶ 1-1 x ∈ 𝒫 V ∖ ∅ | x = 2 ↔ E : dom ⁡ E ⟶ 1-1 x ∈ 𝒫 V | x = 2
6 4 5 mp1i ⊢ G ∈ U → E : dom ⁡ E ⟶ 1-1 x ∈ 𝒫 V ∖ ∅ | x = 2 ↔ E : dom ⁡ E ⟶ 1-1 x ∈ 𝒫 V | x = 2
7 3 6 bitrd ⊢ G ∈ U → G ∈ USGraph ↔ E : dom ⁡ E ⟶ 1-1 x ∈ 𝒫 V | x = 2