Metamath Proof Explorer


Theorem eleei

Description: The forward direction of elee . (Contributed by Scott Fenton, 1-Jul-2013)

Ref Expression
Assertion eleei ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) → 𝐴 : ( 1 ... 𝑁 ) ⟶ ℝ )

Proof

Step Hyp Ref Expression
1 eleenn ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) → 𝑁 ∈ ℕ )
2 elee ( 𝑁 ∈ ℕ → ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ↔ 𝐴 : ( 1 ... 𝑁 ) ⟶ ℝ ) )
3 1 2 syl ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) → ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ↔ 𝐴 : ( 1 ... 𝑁 ) ⟶ ℝ ) )
4 3 ibi ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) → 𝐴 : ( 1 ... 𝑁 ) ⟶ ℝ )