Theorem uspgrupgr

Description: A simple pseudograph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017) (Revised by AV, 15-Oct-2020)

	Ref	Expression
Assertion	uspgrupgr	⊢ ( 𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
3	1 2	isuspgr	⊢ ( 𝐺 ∈ USPGraph → ( 𝐺 ∈ USPGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
4		f1f	⊢ ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
5	3 4	biimtrdi	⊢ ( 𝐺 ∈ USPGraph → ( 𝐺 ∈ USPGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
6	1 2	isupgr	⊢ ( 𝐺 ∈ USPGraph → ( 𝐺 ∈ UPGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
7	5 6	sylibrd	⊢ ( 𝐺 ∈ USPGraph → ( 𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph ) )
8	7	pm2.43i	⊢ ( 𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )