Theorem uspgrunop

Description: The union of two simple pseudographs (with the same vertex set): If <. V , E >. and <. V , F >. are simple pseudographs, then <. V , E u. F >. is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017) (Revised by AV, 16-Oct-2020) (Revised by AV, 24-Oct-2021)

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

Proof

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