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

Theorem iccf 12269
Description: The set of closed intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
iccf [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*

Proof of Theorem iccf
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-icc 12179 . 2 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
21ixxf 12182 1 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
Colors of variables: wff setvar class
Syntax hints:  𝒫 cpw 4156   × cxp 5110  wf 5882  *cxr 10070  cle 10072  [,]cicc 12175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-xr 10075  df-icc 12179
This theorem is referenced by:  lecldbas  21017  ovolficc  23231  ovolficcss  23232  uniiccdif  23340  uniiccvol  23342  dyadmbllem  23361  dyadmbl  23362  opnmbllem  23363  opnmbllem0  33425  mblfinlem1  33426  mblfinlem2  33427
  Copyright terms: Public domain W3C validator