Theorem lo1f

Description: An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016)

	Ref	Expression
Assertion	lo1f	⊢ ( 𝐹 ∈ ≤𝑂(1) → 𝐹 : dom 𝐹 ⟶ ℝ )

Proof

Step	Hyp	Ref	Expression
1		ello1	⊢ ( 𝐹 ∈ ≤𝑂(1) ↔ ( 𝐹 ∈ ( ℝ ↑pm ℝ ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) )
2	1	simplbi	⊢ ( 𝐹 ∈ ≤𝑂(1) → 𝐹 ∈ ( ℝ ↑pm ℝ ) )
3		reex	⊢ ℝ ∈ V
4	3 3	elpm2	⊢ ( 𝐹 ∈ ( ℝ ↑pm ℝ ) ↔ ( 𝐹 : dom 𝐹 ⟶ ℝ ∧ dom 𝐹 ⊆ ℝ ) )
5	4	simplbi	⊢ ( 𝐹 ∈ ( ℝ ↑pm ℝ ) → 𝐹 : dom 𝐹 ⟶ ℝ )
6	2 5	syl	⊢ ( 𝐹 ∈ ≤𝑂(1) → 𝐹 : dom 𝐹 ⟶ ℝ )