Theorem fusgrfupgrfs

Description: A finite simple graph is a finite pseudograph of finite size. (Contributed by AV, 27-Dec-2021)

	Ref	Expression
Hypotheses	fusgrfupgrfs.v	|- V = ( Vtx ` G )
	fusgrfupgrfs.i	|- I = ( iEdg ` G )
Assertion	fusgrfupgrfs	|- ( G e. FinUSGraph -> ( G e. UPGraph /\ V e. Fin /\ I e. Fin ) )

Proof

Step	Hyp	Ref	Expression
1		fusgrfupgrfs.v	|- V = ( Vtx ` G )
2		fusgrfupgrfs.i	|- I = ( iEdg ` G )
3		fusgrusgr	|- ( G e. FinUSGraph -> G e. USGraph )
4		usgrupgr	|- ( G e. USGraph -> G e. UPGraph )
5	3 4	syl	|- ( G e. FinUSGraph -> G e. UPGraph )
6	1	fusgrvtxfi	|- ( G e. FinUSGraph -> V e. Fin )
7		fusgrfis	|- ( G e. FinUSGraph -> ( Edg ` G ) e. Fin )
8		eqid	|- ( Edg ` G ) = ( Edg ` G )
9	2 8	usgredgffibi	|- ( G e. USGraph -> ( ( Edg ` G ) e. Fin <-> I e. Fin ) )
10	3 9	syl	|- ( G e. FinUSGraph -> ( ( Edg ` G ) e. Fin <-> I e. Fin ) )
11	7 10	mpbid	|- ( G e. FinUSGraph -> I e. Fin )
12	5 6 11	3jca	|- ( G e. FinUSGraph -> ( G e. UPGraph /\ V e. Fin /\ I e. Fin ) )