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Theorem ixxssxr 12137
Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)
Hypothesis
Ref Expression
ixx.1 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
Assertion
Ref Expression
ixxssxr (𝐴𝑂𝐵) ⊆ ℝ*
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧
Allowed substitution hints:   𝑂(𝑥,𝑦,𝑧)

Proof of Theorem ixxssxr
StepHypRef Expression
1 df-ov 6613 . . 3 (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
2 ixx.1 . . . . 5 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
32ixxf 12135 . . . 4 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*
4 0elpw 4799 . . . 4 ∅ ∈ 𝒫 ℝ*
53, 4f0cli 6331 . . 3 (𝑂‘⟨𝐴, 𝐵⟩) ∈ 𝒫 ℝ*
61, 5eqeltri 2694 . 2 (𝐴𝑂𝐵) ∈ 𝒫 ℝ*
7 ovex 6638 . . 3 (𝐴𝑂𝐵) ∈ V
87elpw 4141 . 2 ((𝐴𝑂𝐵) ∈ 𝒫 ℝ* ↔ (𝐴𝑂𝐵) ⊆ ℝ*)
96, 8mpbi 220 1 (𝐴𝑂𝐵) ⊆ ℝ*
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  {crab 2911  wss 3559  𝒫 cpw 4135  cop 4159   class class class wbr 4618   × cxp 5077  cfv 5852  (class class class)co 6610  cmpt2 6612  *cxr 10025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-xr 10030
This theorem is referenced by:  iccssxr  12206  iocssxr  12207  icossxr  12208  ioossioobi  39185
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