Theorem usgrun

Description: The union U of two simple graphs G and H with the same vertex set V is a multigraph (not necessarily a simple graph!) with the vertex V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 29-Nov-2020)

	Ref	Expression
Hypotheses	usgrun.g	$$⊢\phi \to G\in \mathrm{USGraph}$$
	usgrun.h	$$⊢\phi \to H\in \mathrm{USGraph}$$
	usgrun.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
	usgrun.f	$$⊢F=\mathrm{iEdg}\left(H\right)$$
	usgrun.vg	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	usgrun.vh	$$⊢\phi \to \mathrm{Vtx}\left(H\right)=V$$
	usgrun.i	$$⊢\phi \to \mathrm{dom}E\cap \mathrm{dom}F=\varnothing$$
	usgrun.u	$$⊢\phi \to U\in W$$
	usgrun.v	$$⊢\phi \to \mathrm{Vtx}\left(U\right)=V$$
	usgrun.un	$$⊢\phi \to \mathrm{iEdg}\left(U\right)=E\cup F$$
Assertion	usgrun	$$⊢\phi \to U\in \mathrm{UMGraph}$$

Proof

Step	Hyp	Ref	Expression
1		usgrun.g	$$⊢\phi \to G\in \mathrm{USGraph}$$
2		usgrun.h	$$⊢\phi \to H\in \mathrm{USGraph}$$
3		usgrun.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
4		usgrun.f	$$⊢F=\mathrm{iEdg}\left(H\right)$$
5		usgrun.vg	$$⊢V=\mathrm{Vtx}\left(G\right)$$
6		usgrun.vh	$$⊢\phi \to \mathrm{Vtx}\left(H\right)=V$$
7		usgrun.i	$$⊢\phi \to \mathrm{dom}E\cap \mathrm{dom}F=\varnothing$$
8		usgrun.u	$$⊢\phi \to U\in W$$
9		usgrun.v	$$⊢\phi \to \mathrm{Vtx}\left(U\right)=V$$
10		usgrun.un	$$⊢\phi \to \mathrm{iEdg}\left(U\right)=E\cup F$$
11		usgrumgr	$$⊢G\in \mathrm{USGraph}\to G\in \mathrm{UMGraph}$$
12	1 11	syl	$$⊢\phi \to G\in \mathrm{UMGraph}$$
13		usgrumgr	$$⊢H\in \mathrm{USGraph}\to H\in \mathrm{UMGraph}$$
14	2 13	syl	$$⊢\phi \to H\in \mathrm{UMGraph}$$
15	12 14 3 4 5 6 7 8 9 10	umgrun	$$⊢\phi \to U\in \mathrm{UMGraph}$$