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Metamath Proof Explorer

Theorem wrdffz

Description: A word is a function from a finite interval of integers. (Contributed by AV, 10-Feb-2021)

Ref Expression
Assertion wrdffz 
|- ( W e. Word S -> W : ( 0 ... ( ( # ` W ) - 1 ) ) --> S )

Proof

Step Hyp Ref Expression
1 wrdf 
 |-  ( W e. Word S -> W : ( 0 ..^ ( # ` W ) ) --> S )
2 lencl 
 |-  ( W e. Word S -> ( # ` W ) e. NN0 )
3 2 nn0zd 
 |-  ( W e. Word S -> ( # ` W ) e. ZZ )
4 fzoval 
 |-  ( ( # ` W ) e. ZZ -> ( 0 ..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) )
5 3 4 syl 
 |-  ( W e. Word S -> ( 0 ..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) )
6 5 feq2d 
 |-  ( W e. Word S -> ( W : ( 0 ..^ ( # ` W ) ) --> S <-> W : ( 0 ... ( ( # ` W ) - 1 ) ) --> S ) )
7 1 6 mpbid 
 |-  ( W e. Word S -> W : ( 0 ... ( ( # ` W ) - 1 ) ) --> S )