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	$$⊢\phi \to G\in \mathrm{USHGraph}$$
	ushgrun.h	$$⊢\phi \to H\in \mathrm{USHGraph}$$
	ushgrun.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
	ushgrun.f	$$⊢F=\mathrm{iEdg}\left(H\right)$$
	ushgrun.vg	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	ushgrun.vh	$$⊢\phi \to \mathrm{Vtx}\left(H\right)=V$$
	ushgrun.i	$$⊢\phi \to \mathrm{dom}E\cap \mathrm{dom}F=\varnothing$$
Assertion	ushgrunop	$$⊢\phi \to ⟨V,E\cup F⟩\in \mathrm{UHGraph}$$

Proof

Step	Hyp	Ref	Expression
1		ushgrun.g	$$⊢\phi \to G\in \mathrm{USHGraph}$$
2		ushgrun.h	$$⊢\phi \to H\in \mathrm{USHGraph}$$
3		ushgrun.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
4		ushgrun.f	$$⊢F=\mathrm{iEdg}\left(H\right)$$
5		ushgrun.vg	$$⊢V=\mathrm{Vtx}\left(G\right)$$
6		ushgrun.vh	$$⊢\phi \to \mathrm{Vtx}\left(H\right)=V$$
7		ushgrun.i	$$⊢\phi \to \mathrm{dom}E\cap \mathrm{dom}F=\varnothing$$
8		ushgruhgr	$$⊢G\in \mathrm{USHGraph}\to G\in \mathrm{UHGraph}$$
9	1 8	syl	$$⊢\phi \to G\in \mathrm{UHGraph}$$
10		ushgruhgr	$$⊢H\in \mathrm{USHGraph}\to H\in \mathrm{UHGraph}$$
11	2 10	syl	$$⊢\phi \to H\in \mathrm{UHGraph}$$
12	9 11 3 4 5 6 7	uhgrunop	$$⊢\phi \to ⟨V,E\cup F⟩\in \mathrm{UHGraph}$$