Theorem iswrdb

Description: A word over an alphabet is a function from an open range of nonnegative integers (of length equal to the length of the word) into the alphabet. (Contributed by Alexander van der Vekens, 30-Jul-2018)

	Ref	Expression
Assertion	iswrdb	⊢ ( 𝑊 ∈ Word 𝑆 ↔ 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		wrdf	⊢ ( 𝑊 ∈ Word 𝑆 → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑆 )
2		iswrdi	⊢ ( 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑆 → 𝑊 ∈ Word 𝑆 )
3	1 2	impbii	⊢ ( 𝑊 ∈ Word 𝑆 ↔ 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑆 )