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 ) ) |
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 ) ) |