Theorem usgrunop

Description: The union of two simple graphs (with the same vertex set): If <. V , E >. and <. V , F >. are simple graphs, then <. V , E u. F >. is a multigraph (not necessarily a simple graph!) - 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)

	Ref	Expression
Hypotheses	usgrun.g	|- ( ph -> G e. USGraph )
	usgrun.h	|- ( ph -> H e. USGraph )
	usgrun.e	|- E = ( iEdg ` G )
	usgrun.f	|- F = ( iEdg ` H )
	usgrun.vg	|- V = ( Vtx ` G )
	usgrun.vh	|- ( ph -> ( Vtx ` H ) = V )
	usgrun.i	|- ( ph -> ( dom E i^i dom F ) = (/) )
Assertion	usgrunop	|- ( ph -> <. V , ( E u. F ) >. e. UMGraph )

Proof

Step	Hyp	Ref	Expression
1		usgrun.g	|- ( ph -> G e. USGraph )
2		usgrun.h	|- ( ph -> H e. USGraph )
3		usgrun.e	|- E = ( iEdg ` G )
4		usgrun.f	|- F = ( iEdg ` H )
5		usgrun.vg	|- V = ( Vtx ` G )
6		usgrun.vh	|- ( ph -> ( Vtx ` H ) = V )
7		usgrun.i	|- ( ph -> ( dom E i^i dom F ) = (/) )
8		usgrumgr	|- ( G e. USGraph -> G e. UMGraph )
9	1 8	syl	|- ( ph -> G e. UMGraph )
10		usgrumgr	|- ( H e. USGraph -> H e. UMGraph )
11	2 10	syl	|- ( ph -> H e. UMGraph )
12	9 11 3 4 5 6 7	umgrunop	|- ( ph -> <. V , ( E u. F ) >. e. UMGraph )