Theorem pfxf

Description: A prefix of a word is a function from a half-open range of nonnegative integers of the same length as the prefix to the set of symbols for the original word. (Contributed by AV, 2-May-2020)

	Ref	Expression
Assertion	pfxf	|- ( ( W e. Word V /\ L e. ( 0 ... ( # ` W ) ) ) -> ( W prefix L ) : ( 0 ..^ L ) --> V )

Proof

Step	Hyp	Ref	Expression
1		pfxmpt	|- ( ( W e. Word V /\ L e. ( 0 ... ( # ` W ) ) ) -> ( W prefix L ) = ( x e. ( 0 ..^ L ) |-> ( W ` x ) ) )
2		simpll	|- ( ( ( W e. Word V /\ L e. ( 0 ... ( # ` W ) ) ) /\ x e. ( 0 ..^ L ) ) -> W e. Word V )
3		elfzuz3	|- ( L e. ( 0 ... ( # ` W ) ) -> ( # ` W ) e. ( ZZ>= ` L ) )
4	3	adantl	|- ( ( W e. Word V /\ L e. ( 0 ... ( # ` W ) ) ) -> ( # ` W ) e. ( ZZ>= ` L ) )
5		fzoss2	|- ( ( # ` W ) e. ( ZZ>= ` L ) -> ( 0 ..^ L ) C_ ( 0 ..^ ( # ` W ) ) )
6	4 5	syl	|- ( ( W e. Word V /\ L e. ( 0 ... ( # ` W ) ) ) -> ( 0 ..^ L ) C_ ( 0 ..^ ( # ` W ) ) )
7	6	sselda	|- ( ( ( W e. Word V /\ L e. ( 0 ... ( # ` W ) ) ) /\ x e. ( 0 ..^ L ) ) -> x e. ( 0 ..^ ( # ` W ) ) )
8		wrdsymbcl	|- ( ( W e. Word V /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` x ) e. V )
9	2 7 8	syl2anc	|- ( ( ( W e. Word V /\ L e. ( 0 ... ( # ` W ) ) ) /\ x e. ( 0 ..^ L ) ) -> ( W ` x ) e. V )
10	1 9	fmpt3d	|- ( ( W e. Word V /\ L e. ( 0 ... ( # ` W ) ) ) -> ( W prefix L ) : ( 0 ..^ L ) --> V )