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 ) |
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 ) |