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Metamath Proof Explorer

Theorem upgrle

Description: An edge of an undirected pseudograph has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015) (Revised by AV, 10-Oct-2020)

Ref Expression
Hypotheses isupgr.v 
|- V = ( Vtx ` G )
isupgr.e 
|- E = ( iEdg ` G )
Assertion upgrle 
|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( # ` ( E ` F ) ) <_ 2 )

Proof

Step Hyp Ref Expression
1 isupgr.v 
 |-  V = ( Vtx ` G )
2 isupgr.e 
 |-  E = ( iEdg ` G )
3 1 2 upgrfn 
 |-  ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
4 3 ffvelcdmda 
 |-  ( ( ( G e. UPGraph /\ E Fn A ) /\ F e. A ) -> ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
5 4 3impa 
 |-  ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
6 fveq2 
 |-  ( x = ( E ` F ) -> ( # ` x ) = ( # ` ( E ` F ) ) )
7 6 breq1d 
 |-  ( x = ( E ` F ) -> ( ( # ` x ) <_ 2 <-> ( # ` ( E ` F ) ) <_ 2 ) )
8 7 elrab 
 |-  ( ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } <-> ( ( E ` F ) e. ( ~P V \ { (/) } ) /\ ( # ` ( E ` F ) ) <_ 2 ) )
9 8 simprbi 
 |-  ( ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } -> ( # ` ( E ` F ) ) <_ 2 )
10 5 9 syl 
 |-  ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( # ` ( E ` F ) ) <_ 2 )