Theorem upgruhgr

Description: An undirected pseudograph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017) (Revised by AV, 10-Oct-2020)

	Ref	Expression
Assertion	upgruhgr	⊢ ( 𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
3	1 2	upgrf	⊢ ( 𝐺 ∈ UPGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
4		ssrab2	⊢ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ⊆ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } )
5		fss	⊢ ( ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ∧ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ⊆ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ) → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) )
6	3 4 5	sylancl	⊢ ( 𝐺 ∈ UPGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) )
7	1 2	isuhgr	⊢ ( 𝐺 ∈ UPGraph → ( 𝐺 ∈ UHGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ) )
8	6 7	mpbird	⊢ ( 𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )