Theorem elbigodm

Description: The domain of a function of order G(x) is a subset of the reals. (Contributed by AV, 18-May-2020)

	Ref	Expression
Assertion	elbigodm	⊢ ( 𝐹 ∈ ( Ο ‘ 𝐺 ) → dom 𝐹 ⊆ ℝ )

Proof

Step	Hyp	Ref	Expression
1		elbigo	⊢ ( 𝐹 ∈ ( Ο ‘ 𝐺 ) ↔ ( 𝐹 ∈ ( ℝ ↑pm ℝ ) ∧ 𝐺 ∈ ( ℝ ↑pm ℝ ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) ( 𝐹 ‘ 𝑦 ) ≤ ( 𝑚 · ( 𝐺 ‘ 𝑦 ) ) ) )
2		reex	⊢ ℝ ∈ V
3	2 2	elpm2	⊢ ( 𝐹 ∈ ( ℝ ↑pm ℝ ) ↔ ( 𝐹 : dom 𝐹 ⟶ ℝ ∧ dom 𝐹 ⊆ ℝ ) )
4	3	simprbi	⊢ ( 𝐹 ∈ ( ℝ ↑pm ℝ ) → dom 𝐹 ⊆ ℝ )
5	4	3ad2ant1	⊢ ( ( 𝐹 ∈ ( ℝ ↑pm ℝ ) ∧ 𝐺 ∈ ( ℝ ↑pm ℝ ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) ( 𝐹 ‘ 𝑦 ) ≤ ( 𝑚 · ( 𝐺 ‘ 𝑦 ) ) ) → dom 𝐹 ⊆ ℝ )
6	1 5	sylbi	⊢ ( 𝐹 ∈ ( Ο ‘ 𝐺 ) → dom 𝐹 ⊆ ℝ )