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	⊢ ( 𝜑 → 𝑁 = ( ♯ ‘ 𝑊 ) )
	wrdfd.w	⊢ ( 𝜑 → 𝑊 ∈ Word 𝑆 )
Assertion	wrdfd	⊢ ( 𝜑 → 𝑊 : ( 0 ..^ 𝑁 ) ⟶ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		wrdfd.n	⊢ ( 𝜑 → 𝑁 = ( ♯ ‘ 𝑊 ) )
2		wrdfd.w	⊢ ( 𝜑 → 𝑊 ∈ Word 𝑆 )
3		wrdf	⊢ ( 𝑊 ∈ Word 𝑆 → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑆 )
4	2 3	syl	⊢ ( 𝜑 → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑆 )
5	1	oveq2d	⊢ ( 𝜑 → ( 0 ..^ 𝑁 ) = ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
6	5	feq2d	⊢ ( 𝜑 → ( 𝑊 : ( 0 ..^ 𝑁 ) ⟶ 𝑆 ↔ 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑆 ) )
7	4 6	mpbird	⊢ ( 𝜑 → 𝑊 : ( 0 ..^ 𝑁 ) ⟶ 𝑆 )