Theorem usgrumgruspgr

Description: A graph is a simple graph iff it is a multigraph and a simple pseudograph. (Contributed by AV, 30-Nov-2020)

	Ref	Expression
Assertion	usgrumgruspgr	⊢ ( 𝐺 ∈ USGraph ↔ ( 𝐺 ∈ UMGraph ∧ 𝐺 ∈ USPGraph ) )

Proof

Step	Hyp	Ref	Expression
1		usgrumgr	⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UMGraph )
2		usgruspgr	⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
3	1 2	jca	⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ UMGraph ∧ 𝐺 ∈ USPGraph ) )
4		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
5		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
6	4 5	uspgrf	⊢ ( 𝐺 ∈ USPGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
7		umgredgss	⊢ ( 𝐺 ∈ UMGraph → ( Edg ‘ 𝐺 ) ⊆ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
8		edgval	⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 )
9		prprrab	⊢ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } = { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 }
10	9	eqcomi	⊢ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } = { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 }
11	7 8 10	3sstr3g	⊢ ( 𝐺 ∈ UMGraph → ran ( iEdg ‘ 𝐺 ) ⊆ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
12		f1ssr	⊢ ( ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ∧ ran ( iEdg ‘ 𝐺 ) ⊆ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
13	6 11 12	syl2anr	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝐺 ∈ USPGraph ) → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
14	4 5	isusgr	⊢ ( 𝐺 ∈ UMGraph → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )
15	14	adantr	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝐺 ∈ USPGraph ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )
16	13 15	mpbird	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝐺 ∈ USPGraph ) → 𝐺 ∈ USGraph )
17	3 16	impbii	⊢ ( 𝐺 ∈ USGraph ↔ ( 𝐺 ∈ UMGraph ∧ 𝐺 ∈ USPGraph ) )