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Metamath Proof Explorer

Theorem isusgrs

Description: The property of being a simple graph, simplified version of isusgr . (Contributed by Alexander van der Vekens, 13-Aug-2017) (Revised by AV, 13-Oct-2020) (Proof shortened by AV, 24-Nov-2020)

Ref Expression
Hypotheses isuspgr.v 
|- V = ( Vtx ` G )
isuspgr.e 
|- E = ( iEdg ` G )
Assertion isusgrs 
|- ( G e. U -> ( G e. USGraph <-> 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 isusgr 
 |-  ( G e. U -> ( G e. USGraph <-> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } ) )
4 prprrab 
 |-  { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } = { x e. ~P V | ( # ` x ) = 2 }
5 f1eq3 
 |-  ( { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } = { x e. ~P V | ( # ` x ) = 2 } -> ( E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } <-> E : dom E -1-1-> { x e. ~P V | ( # ` x ) = 2 } ) )
6 4 5 mp1i 
 |-  ( G e. U -> ( E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } <-> E : dom E -1-1-> { x e. ~P V | ( # ` x ) = 2 } ) )
7 3 6 bitrd 
 |-  ( G e. U -> ( G e. USGraph <-> E : dom E -1-1-> { x e. ~P V | ( # ` x ) = 2 } ) )