Metamath Proof Explorer


Theorem usgrfun

Description: The edge function of a simple graph is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017) (Revised by AV, 13-Oct-2020)

Ref Expression
Assertion usgrfun ( 𝐺 ∈ USGraph → Fun ( iEdg ‘ 𝐺 ) )

Proof

Step Hyp Ref Expression
1 eqid ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2 eqid ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
3 1 2 usgrfs ( 𝐺 ∈ USGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
4 f1fun ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } → Fun ( iEdg ‘ 𝐺 ) )
5 3 4 syl ( 𝐺 ∈ USGraph → Fun ( iEdg ‘ 𝐺 ) )