Metamath Proof Explorer


Theorem usgrf1

Description: The edge function of a simple graph is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017) (Revised by AV, 15-Oct-2020)

Ref Expression
Hypothesis usgrf1o.e
|- E = ( iEdg ` G )
Assertion usgrf1
|- ( G e. USGraph -> E : dom E -1-1-> ran E )

Proof

Step Hyp Ref Expression
1 usgrf1o.e
 |-  E = ( iEdg ` G )
2 1 usgrf1o
 |-  ( G e. USGraph -> E : dom E -1-1-onto-> ran E )
3 f1of1
 |-  ( E : dom E -1-1-onto-> ran E -> E : dom E -1-1-> ran E )
4 2 3 syl
 |-  ( G e. USGraph -> E : dom E -1-1-> ran E )