Theorem elo1mpt

Description: Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014) (Proof shortened by Mario Carneiro, 26-May-2016)

	Ref	Expression
Hypotheses	elo1mpt.1	$$⊢\phi \to A\subseteq ℝ$$
	elo1mpt.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℂ$$
Assertion	elo1mpt	$$⊢\phi \to \left(\left(x\in A⟼B\right)\in 𝑂\left(1\right)\leftrightarrow \exists y\in ℝ\exists m\in ℝ\forall x\in A\left(y\le x\to \left|B\right|\le m\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		elo1mpt.1	$$⊢\phi \to A\subseteq ℝ$$
2		elo1mpt.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℂ$$
3	2	lo1o12	$$⊢\phi \to \left(\left(x\in A⟼B\right)\in 𝑂\left(1\right)\leftrightarrow \left(x\in A⟼\left|B\right|\right)\in \le 𝑂\left(1\right)\right)$$
4	2	abscld	$$⊢\left(\phi \wedge x\in A\right)\to \left|B\right|\in ℝ$$
5	1 4	ello1mpt	$$⊢\phi \to \left(\left(x\in A⟼\left|B\right|\right)\in \le 𝑂\left(1\right)\leftrightarrow \exists y\in ℝ\exists m\in ℝ\forall x\in A\left(y\le x\to \left|B\right|\le m\right)\right)$$
6	3 5	bitrd	$$⊢\phi \to \left(\left(x\in A⟼B\right)\in 𝑂\left(1\right)\leftrightarrow \exists y\in ℝ\exists m\in ℝ\forall x\in A\left(y\le x\to \left|B\right|\le m\right)\right)$$