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Metamath Proof Explorer

Theorem uspgrupgrushgr

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

Ref Expression
Assertion uspgrupgrushgr 
|- ( G e. USPGraph <-> ( G e. UPGraph /\ G e. USHGraph ) )

Proof

Step Hyp Ref Expression
1 uspgrupgr 
 |-  ( G e. USPGraph -> G e. UPGraph )
2 uspgrushgr 
 |-  ( G e. USPGraph -> G e. USHGraph )
3 1 2 jca 
 |-  ( G e. USPGraph -> ( G e. UPGraph /\ G e. USHGraph ) )
4 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
5 eqid 
 |-  ( iEdg ` G ) = ( iEdg ` G )
6 4 5 ushgrf 
 |-  ( G e. USHGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> ( ~P ( Vtx ` G ) \ { (/) } ) )
7 edgval 
 |-  ( Edg ` G ) = ran ( iEdg ` G )
8 upgredgss 
 |-  ( G e. UPGraph -> ( Edg ` G ) C_ { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
9 7 8 eqsstrrid 
 |-  ( G e. UPGraph -> ran ( iEdg ` G ) C_ { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
10 f1ssr 
 |-  ( ( ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> ( ~P ( Vtx ` G ) \ { (/) } ) /\ 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 } )
11 6 9 10 syl2anr 
 |-  ( ( G e. UPGraph /\ G e. USHGraph ) -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
12 4 5 isuspgr 
 |-  ( G e. UPGraph -> ( G e. USPGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
13 12 adantr 
 |-  ( ( G e. UPGraph /\ G e. USHGraph ) -> ( G e. USPGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
14 11 13 mpbird 
 |-  ( ( G e. UPGraph /\ G e. USHGraph ) -> G e. USPGraph )
15 3 14 impbii 
 |-  ( G e. USPGraph <-> ( G e. UPGraph /\ G e. USHGraph ) )