MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iccval Structured version   Visualization version   GIF version

Theorem iccval 12765
Description: Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
iccval ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴[,]𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴𝑥𝑥𝐵)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iccval
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-icc 12733 . 2 [,] = (𝑦 ∈ ℝ*, 𝑧 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑦𝑥𝑥𝑧)})
21ixxval 12734 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴[,]𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴𝑥𝑥𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139   class class class wbr 5057  (class class class)co 7145  *cxr 10662  cle 10664  [,]cicc 12729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-xr 10667  df-icc 12733
This theorem is referenced by:  icc0  12774  iccmax  12800  fzval2  12883  ordtrestixx  21758  cnvordtrestixx  31055  areacirclem5  34867
  Copyright terms: Public domain W3C validator