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 E 1-1 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 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