Theorem usgredgprv

Description: In a simple graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017) (Revised by AV, 16-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

	Ref	Expression
Hypotheses	usgredg2.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
	usgredgprv.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	usgredgprv	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ) → ( ( 𝐸 ‘ 𝑋 ) = { 𝑀 , 𝑁 } → ( 𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ) ) )

Proof

Step	Hyp	Ref	Expression
1		usgredg2.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
2		usgredgprv.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
3		usgrumgr	⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UMGraph )
4	1 2	umgredgprv	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐸 ) → ( ( 𝐸 ‘ 𝑋 ) = { 𝑀 , 𝑁 } → ( 𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ) ) )
5	3 4	sylan	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ) → ( ( 𝐸 ‘ 𝑋 ) = { 𝑀 , 𝑁 } → ( 𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ) ) )