Metamath Proof Explorer


Theorem upgredgss

Description: The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 29-Nov-2020)

Ref Expression
Assertion upgredgss ( 𝐺 ∈ UPGraph → ( 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 upgrf ( 𝐺 ∈ UPGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
5 4 frnd ( 𝐺 ∈ UPGraph → ran ( iEdg ‘ 𝐺 ) ⊆ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
6 1 5 eqsstrid ( 𝐺 ∈ UPGraph → ( Edg ‘ 𝐺 ) ⊆ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )