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 )