Metamath Proof Explorer

Theorem ffzo0hash

Description: The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018)

Ref Expression
Assertion ffzo0hash 
|- ( ( N e. NN0 /\ F Fn ( 0 ..^ N ) ) -> ( # ` F ) = N )

Proof

Step Hyp Ref Expression
1 hashfn 
 |-  ( F Fn ( 0 ..^ N ) -> ( # ` F ) = ( # ` ( 0 ..^ N ) ) )
2 hashfzo0 
 |-  ( N e. NN0 -> ( # ` ( 0 ..^ N ) ) = N )
3 1 2 sylan9eqr 
 |-  ( ( N e. NN0 /\ F Fn ( 0 ..^ N ) ) -> ( # ` F ) = N )