Theorem elo1d

Description: Sufficient condition for 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	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℂ$$
	elo1d.3	$$⊢\phi \to C\in ℝ$$
	elo1d.4	$$⊢\phi \to M\in ℝ$$
	elo1d.5	$$⊢\left(\phi \wedge \left(x\in A\wedge C\le x\right)\right)\to \left|B\right|\le M$$
Assertion	elo1d	$$⊢\phi \to \left(x\in A⟼B\right)\in 𝑂\left(1\right)$$

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		elo1d.4	$$⊢\phi \to M\in ℝ$$
5		elo1d.5	$$⊢\left(\phi \wedge \left(x\in A\wedge C\le x\right)\right)\to \left|B\right|\le M$$
6	2	abscld	$$⊢\left(\phi \wedge x\in A\right)\to \left|B\right|\in ℝ$$
7	1 6 3 4 5	ello1d	$$⊢\phi \to \left(x\in A⟼\left|B\right|\right)\in \le 𝑂\left(1\right)$$
8	2	lo1o12	$$⊢\phi \to \left(\left(x\in A⟼B\right)\in 𝑂\left(1\right)\leftrightarrow \left(x\in A⟼\left|B\right|\right)\in \le 𝑂\left(1\right)\right)$$
9	7 8	mpbird	$$⊢\phi \to \left(x\in A⟼B\right)\in 𝑂\left(1\right)$$