Theorem usgrf1

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

	Ref	Expression
Hypothesis	usgrf1o.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion	usgrf1	⊢ ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸 –1-1→ ran 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		usgrf1o.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
2	1	usgrf1o	⊢ ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 )
3		f1of1	⊢ ( 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 → 𝐸 : dom 𝐸 –1-1→ ran 𝐸 )
4	2 3	syl	⊢ ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸 –1-1→ ran 𝐸 )