Theorem umgredgss

Description: The set of edges of a multigraph is a subset of the set of proper unordered pairs of vertices. (Contributed by AV, 25-Nov-2020)

	Ref	Expression
Assertion	umgredgss	⊢ ( 𝐺 ∈ UMGraph → ( Edg ‘ 𝐺 ) ⊆ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )

Proof

Step	Hyp	Ref	Expression
1		edgval	⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 )
2		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
3		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
4	2 3	umgrf	⊢ ( 𝐺 ∈ UMGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
5	4	frnd	⊢ ( 𝐺 ∈ UMGraph → ran ( iEdg ‘ 𝐺 ) ⊆ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
6	1 5	eqsstrid	⊢ ( 𝐺 ∈ UMGraph → ( Edg ‘ 𝐺 ) ⊆ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )