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Metamath Proof Explorer

Theorem fsequb2

Description: The values of a finite real sequence have an upper bound. (Contributed by NM, 20-Sep-2005) (Proof shortened by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion fsequb2 
|- ( F : ( M ... N ) --> RR -> E. x e. RR A. y e. ran F y <_ x )

Proof

Step Hyp Ref Expression
1 fzfi 
 |-  ( M ... N ) e. Fin
2 ffvelcdm 
 |-  ( ( F : ( M ... N ) --> RR /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR )
3 2 ralrimiva 
 |-  ( F : ( M ... N ) --> RR -> A. k e. ( M ... N ) ( F ` k ) e. RR )
4 fimaxre3 
 |-  ( ( ( M ... N ) e. Fin /\ A. k e. ( M ... N ) ( F ` k ) e. RR ) -> E. x e. RR A. k e. ( M ... N ) ( F ` k ) <_ x )
5 1 3 4 sylancr 
 |-  ( F : ( M ... N ) --> RR -> E. x e. RR A. k e. ( M ... N ) ( F ` k ) <_ x )
6 ffn 
 |-  ( F : ( M ... N ) --> RR -> F Fn ( M ... N ) )
7 breq1 
 |-  ( y = ( F ` k ) -> ( y <_ x <-> ( F ` k ) <_ x ) )
8 7 ralrn 
 |-  ( F Fn ( M ... N ) -> ( A. y e. ran F y <_ x <-> A. k e. ( M ... N ) ( F ` k ) <_ x ) )
9 6 8 syl 
 |-  ( F : ( M ... N ) --> RR -> ( A. y e. ran F y <_ x <-> A. k e. ( M ... N ) ( F ` k ) <_ x ) )
10 9 rexbidv 
 |-  ( F : ( M ... N ) --> RR -> ( E. x e. RR A. y e. ran F y <_ x <-> E. x e. RR A. k e. ( M ... N ) ( F ` k ) <_ x ) )
11 5 10 mpbird 
 |-  ( F : ( M ... N ) --> RR -> E. x e. RR A. y e. ran F y <_ x )