Theorem isuhgrop

Description: The property of being an undirected hypergraph 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, 1-Jan-2020) (Revised by AV, 9-Oct-2020)

	Ref	Expression
Assertion	isuhgrop	⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( ⟨ 𝑉 , 𝐸 ⟩ ∈ UHGraph ↔ 𝐸 : dom 𝐸 ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ) )

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	isuhgr	⊢ ( ⟨ 𝑉 , 𝐸 ⟩ ∈ V → ( ⟨ 𝑉 , 𝐸 ⟩ ∈ UHGraph ↔ ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) : dom ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ⟶ ( 𝒫 ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ∖ { ∅ } ) ) )
5	1 4	mp1i	⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( ⟨ 𝑉 , 𝐸 ⟩ ∈ UHGraph ↔ ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) : dom ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ⟶ ( 𝒫 ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ∖ { ∅ } ) ) )
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	difeq1d	⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( 𝒫 ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ∖ { ∅ } ) = ( 𝒫 𝑉 ∖ { ∅ } ) )
11	6 7 10	feq123d	⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) : dom ( iEdg ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ⟶ ( 𝒫 ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ∖ { ∅ } ) ↔ 𝐸 : dom 𝐸 ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ) )
12	5 11	bitrd	⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( ⟨ 𝑉 , 𝐸 ⟩ ∈ UHGraph ↔ 𝐸 : dom 𝐸 ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ) )