Theorem isumgrs

Description: The simplified property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020)

	Ref	Expression
Hypotheses	isumgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	isumgr.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion	isumgrs	⊢ ( 𝐺 ∈ 𝑈 → ( 𝐺 ∈ UMGraph ↔ 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )

Proof

Step	Hyp	Ref	Expression
1		isumgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isumgr.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
3	1 2	isumgr	⊢ ( 𝐺 ∈ 𝑈 → ( 𝐺 ∈ UMGraph ↔ 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )
4		prprrab	⊢ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } = { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 }
5	4	a1i	⊢ ( 𝐺 ∈ 𝑈 → { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } = { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } )
6	5	feq3d	⊢ ( 𝐺 ∈ 𝑈 → ( 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ↔ 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )
7	3 6	bitrd	⊢ ( 𝐺 ∈ 𝑈 → ( 𝐺 ∈ UMGraph ↔ 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )