Theorem uspgrupgrushgr

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

	Ref	Expression
Assertion	uspgrupgrushgr	⊢ ( 𝐺 ∈ USPGraph ↔ ( 𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) )

Proof

Step	Hyp	Ref	Expression
1		uspgrupgr	⊢ ( 𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
2		uspgrushgr	⊢ ( 𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph )
3	1 2	jca	⊢ ( 𝐺 ∈ USPGraph → ( 𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) )
4		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
5		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
6	4 5	ushgrf	⊢ ( 𝐺 ∈ USHGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) )
7		edgval	⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 )
8		upgredgss	⊢ ( 𝐺 ∈ UPGraph → ( Edg ‘ 𝐺 ) ⊆ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
9	7 8	eqsstrrid	⊢ ( 𝐺 ∈ UPGraph → ran ( iEdg ‘ 𝐺 ) ⊆ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
10		f1ssr	⊢ ( ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∧ ran ( iEdg ‘ 𝐺 ) ⊆ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
11	6 9 10	syl2anr	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
12	4 5	isuspgr	⊢ ( 𝐺 ∈ UPGraph → ( 𝐺 ∈ USPGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
13	12	adantr	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) → ( 𝐺 ∈ USPGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
14	11 13	mpbird	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) → 𝐺 ∈ USPGraph )
15	3 14	impbii	⊢ ( 𝐺 ∈ USPGraph ↔ ( 𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) )