Theorem iswrdsymb

Description: An arbitrary word is a word over an alphabet if all of its symbols belong to the alphabet. (Contributed by AV, 23-Jan-2021)

	Ref	Expression
Assertion	iswrdsymb	|- ( ( W e. Word _V /\ A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) e. V ) -> W e. Word V )

Proof

Step	Hyp	Ref	Expression
1		wrdfn	|- ( W e. Word _V -> W Fn ( 0 ..^ ( # ` W ) ) )
2	1	anim1i	|- ( ( W e. Word _V /\ A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) e. V ) -> ( W Fn ( 0 ..^ ( # ` W ) ) /\ A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) e. V ) )
3		ffnfv	|- ( W : ( 0 ..^ ( # ` W ) ) --> V <-> ( W Fn ( 0 ..^ ( # ` W ) ) /\ A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) e. V ) )
4	2 3	sylibr	|- ( ( W e. Word _V /\ A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) e. V ) -> W : ( 0 ..^ ( # ` W ) ) --> V )
5		iswrdi	|- ( W : ( 0 ..^ ( # ` W ) ) --> V -> W e. Word V )
6	4 5	syl	|- ( ( W e. Word _V /\ A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) e. V ) -> W e. Word V )