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Metamath Proof Explorer

Theorem upgrss

Description: An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015) (Revised by AV, 29-Nov-2020)

Ref Expression
Hypotheses isupgr.v 
|- V = ( Vtx ` G )
isupgr.e 
|- E = ( iEdg ` G )
Assertion upgrss 
|- ( ( G e. UPGraph /\ F e. dom E ) -> ( E ` F ) C_ V )

Proof

Step Hyp Ref Expression
1 isupgr.v 
 |-  V = ( Vtx ` G )
2 isupgr.e 
 |-  E = ( iEdg ` G )
3 ssrab2 
 |-  { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } C_ ( ~P V \ { (/) } )
4 difss 
 |-  ( ~P V \ { (/) } ) C_ ~P V
5 3 4 sstri 
 |-  { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } C_ ~P V
6 1 2 upgrf 
 |-  ( G e. UPGraph -> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
7 6 ffvelcdmda 
 |-  ( ( G e. UPGraph /\ F e. dom E ) -> ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
8 5 7 sselid 
 |-  ( ( G e. UPGraph /\ F e. dom E ) -> ( E ` F ) e. ~P V )
9 8 elpwid 
 |-  ( ( G e. UPGraph /\ F e. dom E ) -> ( E ` F ) C_ V )