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Metamath Proof Explorer

Theorem umgr0e

Description: The empty graph, with vertices but no edges, is a multigraph. (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by AV, 25-Nov-2020)

Ref Expression
Hypotheses umgr0e.g 
|- ( ph -> G e. W )
umgr0e.e 
|- ( ph -> ( iEdg ` G ) = (/) )
Assertion umgr0e 
|- ( ph -> G e. UMGraph )

Proof

Step Hyp Ref Expression
1 umgr0e.g 
 |-  ( ph -> G e. W )
2 umgr0e.e 
 |-  ( ph -> ( iEdg ` G ) = (/) )
3 2 f10d 
 |-  ( ph -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
4 f1f 
 |-  ( ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } -> ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
5 3 4 syl 
 |-  ( ph -> ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
6 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
7 eqid 
 |-  ( iEdg ` G ) = ( iEdg ` G )
8 6 7 isumgr 
 |-  ( G e. W -> ( G e. UMGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
9 1 8 syl 
 |-  ( ph -> ( G e. UMGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
10 5 9 mpbird 
 |-  ( ph -> G e. UMGraph )