Metamath Proof Explorer


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 𝐸1-1→ ran 𝐸 )

Proof

Step Hyp Ref Expression
1 usgrf1o.e 𝐸 = ( iEdg ‘ 𝐺 )
2 1 usgrf1o ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸1-1-onto→ ran 𝐸 )
3 f1of1 ( 𝐸 : dom 𝐸1-1-onto→ ran 𝐸𝐸 : dom 𝐸1-1→ ran 𝐸 )
4 2 3 syl ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸1-1→ ran 𝐸 )