Theorem wrdfd

Description: A word is a zero-based sequence with a recoverable upper limit, deduction version. (Contributed by Thierry Arnoux, 22-Dec-2021)

	Ref	Expression
Hypotheses	wrdfd.n	|- ( ph -> N = ( # ` W ) )
	wrdfd.w	|- ( ph -> W e. Word S )
Assertion	wrdfd	|- ( ph -> W : ( 0 ..^ N ) --> S )

Proof

Step	Hyp	Ref	Expression
1		wrdfd.n	|- ( ph -> N = ( # ` W ) )
2		wrdfd.w	|- ( ph -> W e. Word S )
3		wrdf	|- ( W e. Word S -> W : ( 0 ..^ ( # ` W ) ) --> S )
4	2 3	syl	|- ( ph -> W : ( 0 ..^ ( # ` W ) ) --> S )
5	1	oveq2d	|- ( ph -> ( 0 ..^ N ) = ( 0 ..^ ( # ` W ) ) )
6	5	feq2d	|- ( ph -> ( W : ( 0 ..^ N ) --> S <-> W : ( 0 ..^ ( # ` W ) ) --> S ) )
7	4 6	mpbird	|- ( ph -> W : ( 0 ..^ N ) --> S )