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

Metamath Proof Explorer

Theorem usgredgss

Description: The set of edges of a simple graph is a subset of the set of proper unordered pairs of vertices. (Contributed by AV, 1-Jan-2020) (Revised by AV, 14-Oct-2020)

Ref Expression
Assertion usgredgss 
|- ( G e. USGraph -> ( Edg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )

Proof

Step Hyp Ref Expression
1 edgval 
 |-  ( Edg ` G ) = ran ( iEdg ` G )
2 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
3 eqid 
 |-  ( iEdg ` G ) = ( iEdg ` G )
4 2 3 usgrfs 
 |-  ( G e. USGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
5 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 } )
6 frn 
 |-  ( ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } -> ran ( iEdg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
7 4 5 6 3syl 
 |-  ( G e. USGraph -> ran ( iEdg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
8 1 7 eqsstrid 
 |-  ( G e. USGraph -> ( Edg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )