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Theorem iooval 12757
Description: Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
iooval ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥𝑥 < 𝐵)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iooval
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioo 12737 . 2 (,) = (𝑦 ∈ ℝ*, 𝑧 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑦 < 𝑥𝑥 < 𝑧)})
21ixxval 12741 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥𝑥 < 𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {crab 3137   class class class wbr 5053  (class class class)co 7146  *cxr 10668   < clt 10669  (,)cioo 12733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7452  ax-cnex 10587  ax-resscn 10588
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-xr 10673  df-ioo 12737
This theorem is referenced by:  ioo0  12758  iooval2  12766  ioof  12832  relowlssretop  34694
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