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Theorem iocval 12417
Description: Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
iocval ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴(,]𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥𝑥𝐵)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iocval
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioc 12385 . 2 (,] = (𝑦 ∈ ℝ*, 𝑧 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑦 < 𝑥𝑥𝑧)})
21ixxval 12388 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴(,]𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥𝑥𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {crab 3065   class class class wbr 4787  (class class class)co 6796  *cxr 10279   < clt 10280  cle 10281  (,]cioc 12381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-xr 10284  df-ioc 12385
This theorem is referenced by:  ioc0  12427  orvclteel  30874
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