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 ∈ Word 𝑆 → 𝑊 : ( 0 ... ( ( ♯ ‘ 𝑊 ) − 1 ) ) ⟶ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		wrdf	⊢ ( 𝑊 ∈ Word 𝑆 → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑆 )
2		lencl	⊢ ( 𝑊 ∈ Word 𝑆 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 )
3	2	nn0zd	⊢ ( 𝑊 ∈ Word 𝑆 → ( ♯ ‘ 𝑊 ) ∈ ℤ )
4		fzoval	⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℤ → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = ( 0 ... ( ( ♯ ‘ 𝑊 ) − 1 ) ) )
5	3 4	syl	⊢ ( 𝑊 ∈ Word 𝑆 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = ( 0 ... ( ( ♯ ‘ 𝑊 ) − 1 ) ) )
6	5	feq2d	⊢ ( 𝑊 ∈ Word 𝑆 → ( 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑆 ↔ 𝑊 : ( 0 ... ( ( ♯ ‘ 𝑊 ) − 1 ) ) ⟶ 𝑆 ) )
7	1 6	mpbid	⊢ ( 𝑊 ∈ Word 𝑆 → 𝑊 : ( 0 ... ( ( ♯ ‘ 𝑊 ) − 1 ) ) ⟶ 𝑆 )