Theorem elo1

Description: Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014)

	Ref	Expression
Assertion	elo1	⊢ ( 𝐹 ∈ 𝑂(1) ↔ ( 𝐹 ∈ ( ℂ ↑pm ℝ ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 𝑚 ) )

Proof

Step	Hyp	Ref	Expression
1		dmeq	⊢ ( 𝑓 = 𝐹 → dom 𝑓 = dom 𝐹 )
2	1	ineq1d	⊢ ( 𝑓 = 𝐹 → ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) = ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) )
3		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
4	3	fveq2d	⊢ ( 𝑓 = 𝐹 → ( abs ‘ ( 𝑓 ‘ 𝑦 ) ) = ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) )
5	4	breq1d	⊢ ( 𝑓 = 𝐹 → ( ( abs ‘ ( 𝑓 ‘ 𝑦 ) ) ≤ 𝑚 ↔ ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 𝑚 ) )
6	2 5	raleqbidv	⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( abs ‘ ( 𝑓 ‘ 𝑦 ) ) ≤ 𝑚 ↔ ∀ 𝑦 ∈ ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 𝑚 ) )
7	6	2rexbidv	⊢ ( 𝑓 = 𝐹 → ( ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( abs ‘ ( 𝑓 ‘ 𝑦 ) ) ≤ 𝑚 ↔ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 𝑚 ) )
8		df-o1	⊢ 𝑂(1) = { 𝑓 ∈ ( ℂ ↑pm ℝ ) ∣ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( abs ‘ ( 𝑓 ‘ 𝑦 ) ) ≤ 𝑚 }
9	7 8	elrab2	⊢ ( 𝐹 ∈ 𝑂(1) ↔ ( 𝐹 ∈ ( ℂ ↑pm ℝ ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 𝑚 ) )