Theorem elo1mpt2

Description: Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 12-May-2016) (Proof shortened by Mario Carneiro, 26-May-2016)

	Ref	Expression
Hypotheses	elo1mpt.1	$$⊢\phi \to A\subseteq ℝ$$
	elo1mpt.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℂ$$
	elo1d.3	$$⊢\phi \to C\in ℝ$$
Assertion	elo1mpt2	$$⊢\phi \to \left(\left(x\in A⟼B\right)\in 𝑂\left(1\right)\leftrightarrow \exists y\in \left[C,\mathrm{+\infty }\right)\exists m\in ℝ\forall x\in A\left(y\le x\to \left|B\right|\le m\right)\right)$$

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		elo1d.3	$$⊢\phi \to C\in ℝ$$
4	2	lo1o12	$$⊢\phi \to \left(\left(x\in A⟼B\right)\in 𝑂\left(1\right)\leftrightarrow \left(x\in A⟼\left|B\right|\right)\in \le 𝑂\left(1\right)\right)$$
5	2	abscld	$$⊢\left(\phi \wedge x\in A\right)\to \left|B\right|\in ℝ$$
6	1 5 3	ello1mpt2	$$⊢\phi \to \left(\left(x\in A⟼\left|B\right|\right)\in \le 𝑂\left(1\right)\leftrightarrow \exists y\in \left[C,\mathrm{+\infty }\right)\exists m\in ℝ\forall x\in A\left(y\le x\to \left|B\right|\le m\right)\right)$$
7	4 6	bitrd	$$⊢\phi \to \left(\left(x\in A⟼B\right)\in 𝑂\left(1\right)\leftrightarrow \exists y\in \left[C,\mathrm{+\infty }\right)\exists m\in ℝ\forall x\in A\left(y\le x\to \left|B\right|\le m\right)\right)$$