Theorem usgrf1o

Description: The edge function of a simple graph is a bijective function onto its range. (Contributed by Alexander van der Vekens, 18-Nov-2017) (Revised by AV, 15-Oct-2020)

	Ref	Expression
Hypothesis	usgrf1o.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion	usgrf1o	⊢ ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		usgrf1o.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
2		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
3	2 1	usgrfs	⊢ ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸 –1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
4		f1f1orn	⊢ ( 𝐸 : dom 𝐸 –1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } → 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 )
5	3 4	syl	⊢ ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 )