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Metamath Proof Explorer

Theorem isusgrop

Description: The property of being an undirected simple graph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of Bollobas p. 1. (Contributed by AV, 30-Nov-2020)

Ref Expression
Assertion isusgrop ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( ⟨ 𝑉 , 𝐸 ⟩ ∈ USGraph ↔ 𝐸 : dom 𝐸 –1-1→ { 𝑝 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑝 ) = 2 } ) )

Proof

Step Hyp Ref Expression
1 opex ⊢ ⟨ 𝑉 , 𝐸 ⟩ ∈ V
2 eqid ⊢ ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ )
3 eqid ⊢ ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ )
4 2 3 isusgrs ⊢ ( ⟨ 𝑉 , 𝐸 ⟩ ∈ V → ( ⟨ 𝑉 , 𝐸 ⟩ ∈ USGraph ↔ ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) : dom ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) –1-1→ { 𝑝 ∈ 𝒫 ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ∣ ( ♯ ‘ 𝑝 ) = 2 } ) )
5 1 4 mp1i ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( ⟨ 𝑉 , 𝐸 ⟩ ∈ USGraph ↔ ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) : dom ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) –1-1→ { 𝑝 ∈ 𝒫 ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ∣ ( ♯ ‘ 𝑝 ) = 2 } ) )
6 opiedgfv ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = 𝐸 )
7 6 dmeqd ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → dom ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = dom 𝐸 )
8 opvtxfv ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = 𝑉 )
9 8 pweqd ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → 𝒫 ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = 𝒫 𝑉 )
10 9 rabeqdv ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → { 𝑝 ∈ 𝒫 ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ∣ ( ♯ ‘ 𝑝 ) = 2 } = { 𝑝 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑝 ) = 2 } )
11 6 7 10 f1eq123d ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) : dom ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) –1-1→ { 𝑝 ∈ 𝒫 ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ∣ ( ♯ ‘ 𝑝 ) = 2 } ↔ 𝐸 : dom 𝐸 –1-1→ { 𝑝 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑝 ) = 2 } ) )
12 5 11 bitrd ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( ⟨ 𝑉 , 𝐸 ⟩ ∈ USGraph ↔ 𝐸 : dom 𝐸 –1-1→ { 𝑝 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑝 ) = 2 } ) )