Metamath Proof Explorer


Theorem usgredg2

Description: The value of the "edge function" of a simple graph is a set containing two elements (the vertices the corresponding edge is connecting). (Contributed by Alexander van der Vekens, 11-Aug-2017) (Revised by AV, 16-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

Ref Expression
Hypothesis usgredg2.e
|- E = ( iEdg ` G )
Assertion usgredg2
|- ( ( G e. USGraph /\ X e. dom E ) -> ( # ` ( E ` X ) ) = 2 )

Proof

Step Hyp Ref Expression
1 usgredg2.e
 |-  E = ( iEdg ` G )
2 usgrumgr
 |-  ( G e. USGraph -> G e. UMGraph )
3 eqid
 |-  ( Vtx ` G ) = ( Vtx ` G )
4 3 1 umgredg2
 |-  ( ( G e. UMGraph /\ X e. dom E ) -> ( # ` ( E ` X ) ) = 2 )
5 2 4 sylan
 |-  ( ( G e. USGraph /\ X e. dom E ) -> ( # ` ( E ` X ) ) = 2 )