This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem isumgrs

Description: The simplified property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020)

Ref Expression
Hypotheses isumgr.v 
|- V = ( Vtx ` G )
isumgr.e 
|- E = ( iEdg ` G )
Assertion isumgrs 
|- ( G e. U -> ( G e. UMGraph <-> E : dom E --> { x e. ~P V | ( # ` x ) = 2 } ) )

Proof

Step Hyp Ref Expression
1 isumgr.v 
 |-  V = ( Vtx ` G )
2 isumgr.e 
 |-  E = ( iEdg ` G )
3 1 2 isumgr 
 |-  ( G e. U -> ( G e. UMGraph <-> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } ) )
4 prprrab 
 |-  { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } = { x e. ~P V | ( # ` x ) = 2 }
5 4 a1i 
 |-  ( G e. U -> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } = { x e. ~P V | ( # ` x ) = 2 } )
6 5 feq3d 
 |-  ( G e. U -> ( E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } <-> E : dom E --> { x e. ~P V | ( # ` x ) = 2 } ) )
7 3 6 bitrd 
 |-  ( G e. U -> ( G e. UMGraph <-> E : dom E --> { x e. ~P V | ( # ` x ) = 2 } ) )