Metamath Proof Explorer


Theorem ffz0hash

Description: The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018) (Proof shortened by AV, 11-Apr-2021)

Ref Expression
Assertion ffz0hash
|- ( ( N e. NN0 /\ F : ( 0 ... N ) --> B ) -> ( # ` F ) = ( N + 1 ) )

Proof

Step Hyp Ref Expression
1 ffn
 |-  ( F : ( 0 ... N ) --> B -> F Fn ( 0 ... N ) )
2 fnfz0hash
 |-  ( ( N e. NN0 /\ F Fn ( 0 ... N ) ) -> ( # ` F ) = ( N + 1 ) )
3 1 2 sylan2
 |-  ( ( N e. NN0 /\ F : ( 0 ... N ) --> B ) -> ( # ` F ) = ( N + 1 ) )