Metamath Proof Explorer

Theorem fnfz0hash

Description: The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018)

Ref Expression
Assertion fnfz0hash 
|- ( ( N e. NN0 /\ F Fn ( 0 ... N ) ) -> ( # ` F ) = ( N + 1 ) )

Proof

Step Hyp Ref Expression
1 hashfn 
 |-  ( F Fn ( 0 ... N ) -> ( # ` F ) = ( # ` ( 0 ... N ) ) )
2 hashfz0 
 |-  ( N e. NN0 -> ( # ` ( 0 ... N ) ) = ( N + 1 ) )
3 1 2 sylan9eqr 
 |-  ( ( N e. NN0 /\ F Fn ( 0 ... N ) ) -> ( # ` F ) = ( N + 1 ) )