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Theorem ixxf 12056
 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.)
Hypothesis
Ref Expression
ixx.1 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
Assertion
Ref Expression
ixxf 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝑆,𝑦,𝑧
Allowed substitution hints:   𝑂(𝑥,𝑦,𝑧)

Proof of Theorem ixxf
StepHypRef Expression
1 ssrab2 3650 . . . 4 {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)} ⊆ ℝ*
2 xrex 11705 . . . . 5 * ∈ V
32elpw2 4755 . . . 4 ({𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)} ∈ 𝒫 ℝ* ↔ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)} ⊆ ℝ*)
41, 3mpbir 220 . . 3 {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)} ∈ 𝒫 ℝ*
54rgen2w 2909 . 2 𝑥 ∈ ℝ*𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)} ∈ 𝒫 ℝ*
6 ixx.1 . . 3 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
76fmpt2 7126 . 2 (∀𝑥 ∈ ℝ*𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)} ∈ 𝒫 ℝ*𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*)
85, 7mpbi 219 1 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583   × cxp 5036  ⟶wf 5800   ↦ cmpt2 6551  ℝ*cxr 9952 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-xr 9957 This theorem is referenced by:  ixxex  12057  ixxssxr  12058  elixx3g  12059  ndmioo  12073  iccf  12143  iocpnfordt  20829  icomnfordt  20830  tpr2rico  29286  icoreresf  32376  icoreelrn  32385  relowlpssretop  32388
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