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Theorem ixxssxr 12436
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 6881 . . 3 (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
2 ixx.1 . . . . 5 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
32ixxf 12434 . . . 4 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*
4 0elpw 5026 . . . 4 ∅ ∈ 𝒫 ℝ*
53, 4f0cli 6596 . . 3 (𝑂‘⟨𝐴, 𝐵⟩) ∈ 𝒫 ℝ*
61, 5eqeltri 2874 . 2 (𝐴𝑂𝐵) ∈ 𝒫 ℝ*
7 ovex 6910 . . 3 (𝐴𝑂𝐵) ∈ V
87elpw 4355 . 2 ((𝐴𝑂𝐵) ∈ 𝒫 ℝ* ↔ (𝐴𝑂𝐵) ⊆ ℝ*)
96, 8mpbi 222 1 (𝐴𝑂𝐵) ⊆ ℝ*
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  wcel 2157  {crab 3093  wss 3769  𝒫 cpw 4349  cop 4374   class class class wbr 4843   × cxp 5310  cfv 6101  (class class class)co 6878  cmpt2 6880  *cxr 10362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-xr 10367
This theorem is referenced by:  iccssxr  12505  iocssxr  12506  icossxr  12507  ioossioobi  40488
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