Theorem ushgrunop

Description: The union of two (undirected) simple hypergraphs (with the same vertex set) represented as ordered pair: If <. V , E >. and <. V , F >. are simple hypergraphs, then <. V , E u. F >. is a (not necessarily simple) hypergraph - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (Contributed by AV, 29-Nov-2020) (Revised by AV, 24-Oct-2021)

	Ref	Expression
Hypotheses	ushgrun.g	⊢ ( 𝜑 → 𝐺 ∈ USHGraph )
	ushgrun.h	⊢ ( 𝜑 → 𝐻 ∈ USHGraph )
	ushgrun.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
	ushgrun.f	⊢ 𝐹 = ( iEdg ‘ 𝐻 )
	ushgrun.vg	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	ushgrun.vh	⊢ ( 𝜑 → ( Vtx ‘ 𝐻 ) = 𝑉 )
	ushgrun.i	⊢ ( 𝜑 → ( dom 𝐸 ∩ dom 𝐹 ) = ∅ )
Assertion	ushgrunop	⊢ ( 𝜑 → ⟨ 𝑉 , ( 𝐸 ∪ 𝐹 ) ⟩ ∈ UHGraph )

Proof

Step	Hyp	Ref	Expression
1		ushgrun.g	⊢ ( 𝜑 → 𝐺 ∈ USHGraph )
2		ushgrun.h	⊢ ( 𝜑 → 𝐻 ∈ USHGraph )
3		ushgrun.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
4		ushgrun.f	⊢ 𝐹 = ( iEdg ‘ 𝐻 )
5		ushgrun.vg	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
6		ushgrun.vh	⊢ ( 𝜑 → ( Vtx ‘ 𝐻 ) = 𝑉 )
7		ushgrun.i	⊢ ( 𝜑 → ( dom 𝐸 ∩ dom 𝐹 ) = ∅ )
8		ushgruhgr	⊢ ( 𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph )
9	1 8	syl	⊢ ( 𝜑 → 𝐺 ∈ UHGraph )
10		ushgruhgr	⊢ ( 𝐻 ∈ USHGraph → 𝐻 ∈ UHGraph )
11	2 10	syl	⊢ ( 𝜑 → 𝐻 ∈ UHGraph )
12	9 11 3 4 5 6 7	uhgrunop	⊢ ( 𝜑 → ⟨ 𝑉 , ( 𝐸 ∪ 𝐹 ) ⟩ ∈ UHGraph )