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 ∈ FinUSGraph → ( 𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin ) )

Proof

Step	Hyp	Ref	Expression
1		fusgrfupgrfs.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		fusgrfupgrfs.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3		fusgrusgr	⊢ ( 𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
4		usgrupgr	⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UPGraph )
5	3 4	syl	⊢ ( 𝐺 ∈ FinUSGraph → 𝐺 ∈ UPGraph )
6	1	fusgrvtxfi	⊢ ( 𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin )
7		fusgrfis	⊢ ( 𝐺 ∈ FinUSGraph → ( Edg ‘ 𝐺 ) ∈ Fin )
8		eqid	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
9	2 8	usgredgffibi	⊢ ( 𝐺 ∈ USGraph → ( ( Edg ‘ 𝐺 ) ∈ Fin ↔ 𝐼 ∈ Fin ) )
10	3 9	syl	⊢ ( 𝐺 ∈ FinUSGraph → ( ( Edg ‘ 𝐺 ) ∈ Fin ↔ 𝐼 ∈ Fin ) )
11	7 10	mpbid	⊢ ( 𝐺 ∈ FinUSGraph → 𝐼 ∈ Fin )
12	5 6 11	3jca	⊢ ( 𝐺 ∈ FinUSGraph → ( 𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin ) )