Theorem usgrf

Description: The edge function of a simple graph is a one-to-one function into the set of proper unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017) (Revised by AV, 13-Oct-2020)

	Ref	Expression
Hypotheses	isuspgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	isuspgr.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion	usgrf	⊢ ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸 –1-1→ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )

Proof

Step	Hyp	Ref	Expression
1		isuspgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isuspgr.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
3	1 2	isusgr	⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ USGraph ↔ 𝐸 : dom 𝐸 –1-1→ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )
4	3	ibi	⊢ ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸 –1-1→ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )