Theorem ixxf

Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007) (Revised by Mario Carneiro, 16-Nov-2013)

	Ref	Expression
Hypothesis	ixx.1	$$⊢O=\left(x\in {ℝ}^{*},y\in {ℝ}^{*}⟼\left\{z\in {ℝ}^{*}|\left(xRz\wedge zSy\right)\right\}\right)$$
Assertion	ixxf	$$⊢O:{ℝ}^{*}\times {ℝ}^{*}⟶𝒫{ℝ}^{*}$$

Proof

Step	Hyp	Ref	Expression
1		ixx.1	$$⊢O=\left(x\in {ℝ}^{*},y\in {ℝ}^{*}⟼\left\{z\in {ℝ}^{*}|\left(xRz\wedge zSy\right)\right\}\right)$$
2		xrex	$$⊢{ℝ}^{*}\in \mathrm{V}$$
3		ssrab2	$$⊢\left\{z\in {ℝ}^{*}|\left(xRz\wedge zSy\right)\right\}\subseteq {ℝ}^{*}$$
4	2 3	elpwi2	$$⊢\left\{z\in {ℝ}^{*}|\left(xRz\wedge zSy\right)\right\}\in 𝒫{ℝ}^{*}$$
5	4	rgen2w	$$⊢\forall x\in {ℝ}^{*}\forall y\in {ℝ}^{*}\left\{z\in {ℝ}^{*}|\left(xRz\wedge zSy\right)\right\}\in 𝒫{ℝ}^{*}$$
6	1	fmpo	$$⊢\forall x\in {ℝ}^{*}\forall y\in {ℝ}^{*}\left\{z\in {ℝ}^{*}|\left(xRz\wedge zSy\right)\right\}\in 𝒫{ℝ}^{*}\leftrightarrow O:{ℝ}^{*}\times {ℝ}^{*}⟶𝒫{ℝ}^{*}$$
7	5 6	mpbi	$$⊢O:{ℝ}^{*}\times {ℝ}^{*}⟶𝒫{ℝ}^{*}$$