Theorem ioof

Description: The set of open intervals of extended reals maps to subsets of reals. (Contributed by NM, 7-Feb-2007) (Revised by Mario Carneiro, 16-Nov-2013)

	Ref	Expression
Assertion	ioof	$$⊢\left(.\right):{ℝ}^{*}\times {ℝ}^{*}⟶𝒫ℝ$$

Proof

Step	Hyp	Ref	Expression
1		iooval	$$⊢\left(x\in {ℝ}^{*}\wedge y\in {ℝ}^{*}\right)\to \left(x,y\right)=\left\{z\in {ℝ}^{*}|\left(x<z\wedge z<y\right)\right\}$$
2		ioossre	$$⊢\left(x,y\right)\subseteq ℝ$$
3		ovex	$$⊢\left(x,y\right)\in \mathrm{V}$$
4	3	elpw	$$⊢\left(x,y\right)\in 𝒫ℝ\leftrightarrow \left(x,y\right)\subseteq ℝ$$
5	2 4	mpbir	$$⊢\left(x,y\right)\in 𝒫ℝ$$
6	1 5	eqeltrrdi	$$⊢\left(x\in {ℝ}^{*}\wedge y\in {ℝ}^{*}\right)\to \left\{z\in {ℝ}^{*}|\left(x<z\wedge z<y\right)\right\}\in 𝒫ℝ$$
7	6	rgen2	$$⊢\forall x\in {ℝ}^{*}\forall y\in {ℝ}^{*}\left\{z\in {ℝ}^{*}|\left(x<z\wedge z<y\right)\right\}\in 𝒫ℝ$$
8		df-ioo	$$⊢\left(.\right)=\left(x\in {ℝ}^{*},y\in {ℝ}^{*}⟼\left\{z\in {ℝ}^{*}|\left(x<z\wedge z<y\right)\right\}\right)$$
9	8	fmpo	$$⊢\forall x\in {ℝ}^{*}\forall y\in {ℝ}^{*}\left\{z\in {ℝ}^{*}|\left(x<z\wedge z<y\right)\right\}\in 𝒫ℝ\leftrightarrow \left(.\right):{ℝ}^{*}\times {ℝ}^{*}⟶𝒫ℝ$$
10	7 9	mpbi	$$⊢\left(.\right):{ℝ}^{*}\times {ℝ}^{*}⟶𝒫ℝ$$