Theorem ushgrun

Description: The union U of two (undirected) simple hypergraphs G and H with the same vertex set V is a (not necessarily simple) hypergraph with the vertex V and the union ( E u. F ) of the (indexed) edges. (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	$$⊢\phi \to \mathrm{dom}E\cap \mathrm{dom}F=\varnothing$$
	ushgrun.u	$$⊢\phi \to U\in W$$
	ushgrun.v	$$⊢\phi \to \mathrm{Vtx}\left(U\right)=V$$
	ushgrun.un	$$⊢\phi \to \mathrm{iEdg}\left(U\right)=E\cup F$$
Assertion	ushgrun	$$⊢\phi \to U\in \mathrm{UHGraph}$$

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		ushgrun.u	$$⊢\phi \to U\in W$$
9		ushgrun.v	$$⊢\phi \to \mathrm{Vtx}\left(U\right)=V$$
10		ushgrun.un	$$⊢\phi \to \mathrm{iEdg}\left(U\right)=E\cup F$$
11		ushgruhgr	$$⊢G\in \mathrm{USHGraph}\to G\in \mathrm{UHGraph}$$
12	1 11	syl	$$⊢\phi \to G\in \mathrm{UHGraph}$$
13		ushgruhgr	$$⊢H\in \mathrm{USHGraph}\to H\in \mathrm{UHGraph}$$
14	2 13	syl	$$⊢\phi \to H\in \mathrm{UHGraph}$$
15	12 14 3 4 5 6 7 8 9 10	uhgrun	$$⊢\phi \to U\in \mathrm{UHGraph}$$