Metamath Proof Explorer


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 𝐸1-1-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 𝐸1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
4 f1f1orn ( 𝐸 : dom 𝐸1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } → 𝐸 : dom 𝐸1-1-onto→ ran 𝐸 )
5 3 4 syl ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸1-1-onto→ ran 𝐸 )