Metamath Proof Explorer


Theorem uspgrf

Description: The edge function of a simple pseudograph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017) (Revised by AV, 13-Oct-2020)

Ref Expression
Hypotheses isuspgr.v
|- V = ( Vtx ` G )
isuspgr.e
|- E = ( iEdg ` G )
Assertion uspgrf
|- ( G e. USPGraph -> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )

Proof

Step Hyp Ref Expression
1 isuspgr.v
 |-  V = ( Vtx ` G )
2 isuspgr.e
 |-  E = ( iEdg ` G )
3 1 2 isuspgr
 |-  ( G e. USPGraph -> ( G e. USPGraph <-> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
4 3 ibi
 |-  ( G e. USPGraph -> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )