Theorem upgredgssspr

Description: The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021)

	Ref	Expression
Assertion	upgredgssspr	⊢ ( 𝐺 ∈ UPGraph → ( Edg ‘ 𝐺 ) ⊆ ( Pairs ‘ ( Vtx ‘ 𝐺 ) ) )

Proof

Step	Hyp	Ref	Expression
1		upgredgss	⊢ ( 𝐺 ∈ UPGraph → ( Edg ‘ 𝐺 ) ⊆ { 𝑝 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑝 ) ≤ 2 } )
2		fvex	⊢ ( Vtx ‘ 𝐺 ) ∈ V
3		sprvalpwle2	⊢ ( ( Vtx ‘ 𝐺 ) ∈ V → ( Pairs ‘ ( Vtx ‘ 𝐺 ) ) = { 𝑝 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑝 ) ≤ 2 } )
4	2 3	ax-mp	⊢ ( Pairs ‘ ( Vtx ‘ 𝐺 ) ) = { 𝑝 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑝 ) ≤ 2 }
5	1 4	sseqtrrdi	⊢ ( 𝐺 ∈ UPGraph → ( Edg ‘ 𝐺 ) ⊆ ( Pairs ‘ ( Vtx ‘ 𝐺 ) ) )