Theorem pfxfn

Description: Value of the prefix extractor as function with domain. (Contributed by AV, 2-May-2020)

	Ref	Expression
Assertion	pfxfn	⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( 𝑆 prefix 𝐿 ) Fn ( 0 ..^ 𝐿 ) )

Proof

Step	Hyp	Ref	Expression
1		pfxf	⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( 𝑆 prefix 𝐿 ) : ( 0 ..^ 𝐿 ) ⟶ 𝑉 )
2	1	ffnd	⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( 𝑆 prefix 𝐿 ) Fn ( 0 ..^ 𝐿 ) )