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Metamath Proof Explorer

Theorem usgrumgruspgr

Description: A graph is a simple graph iff it is a multigraph and a simple pseudograph. (Contributed by AV, 30-Nov-2020)

Ref Expression
Assertion usgrumgruspgr 
|- ( G e. USGraph <-> ( G e. UMGraph /\ G e. USPGraph ) )

Proof

Step Hyp Ref Expression
1 usgrumgr 
 |-  ( G e. USGraph -> G e. UMGraph )
2 usgruspgr 
 |-  ( G e. USGraph -> G e. USPGraph )
3 1 2 jca 
 |-  ( G e. USGraph -> ( G e. UMGraph /\ G e. USPGraph ) )
4 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
5 eqid 
 |-  ( iEdg ` G ) = ( iEdg ` G )
6 4 5 uspgrf 
 |-  ( G e. USPGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
7 umgredgss 
 |-  ( G e. UMGraph -> ( Edg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
8 edgval 
 |-  ( Edg ` G ) = ran ( iEdg ` G )
9 prprrab 
 |-  { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } = { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 }
10 9 eqcomi 
 |-  { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } = { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 }
11 7 8 10 3sstr3g 
 |-  ( G e. UMGraph -> ran ( iEdg ` G ) C_ { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
12 f1ssr 
 |-  ( ( ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } /\ ran ( iEdg ` G ) C_ { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
13 6 11 12 syl2anr 
 |-  ( ( G e. UMGraph /\ G e. USPGraph ) -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
14 4 5 isusgr 
 |-  ( G e. UMGraph -> ( G e. USGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
15 14 adantr 
 |-  ( ( G e. UMGraph /\ G e. USPGraph ) -> ( G e. USGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
16 13 15 mpbird 
 |-  ( ( G e. UMGraph /\ G e. USPGraph ) -> G e. USGraph )
17 3 16 impbii 
 |-  ( G e. USGraph <-> ( G e. UMGraph /\ G e. USPGraph ) )