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 𝐸 –1-1-onto→ ran 𝐸 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrf1o.e | ⊢ 𝐸 = ( iEdg ‘ 𝐺 ) | |
2 | eqid | ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) | |
3 | 2 1 | usgrfs | ⊢ ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸 –1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) |
4 | f1f1orn | ⊢ ( 𝐸 : dom 𝐸 –1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } → 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 ) | |
5 | 3 4 | syl | ⊢ ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 ) |