Theorem upgracycumgr

Description: An acyclic pseudograph is a multigraph. (Contributed by BTernaryTau, 15-Oct-2023)

	Ref	Expression
Assertion	upgracycumgr	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph ) → 𝐺 ∈ UMGraph )

Proof

Step	Hyp	Ref	Expression
1		upgruhgr	⊢ ( 𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
2	1	anim1ci	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph ) → ( 𝐺 ∈ AcyclicGraph ∧ 𝐺 ∈ UHGraph ) )
3		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
4		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
5	3 4	acycgrislfgr	⊢ ( ( 𝐺 ∈ AcyclicGraph ∧ 𝐺 ∈ UHGraph ) → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } )
6	2 5	syl	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph ) → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } )
7	3 4	umgrislfupgr	⊢ ( 𝐺 ∈ UMGraph ↔ ( 𝐺 ∈ UPGraph ∧ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) )
8	7	biimpri	⊢ ( ( 𝐺 ∈ UPGraph ∧ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) → 𝐺 ∈ UMGraph )
9	6 8	syldan	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph ) → 𝐺 ∈ UMGraph )