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Theorem dmico 40287
Description: The domain of the closed-below, open-above interval function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Assertion
Ref Expression
dmico dom [,) = (ℝ* × ℝ*)

Proof of Theorem dmico
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ico 12366 . . 3 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
21ixxf 12370 . 2 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
32fdmi 6205 1 dom [,) = (ℝ* × ℝ*)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1624  𝒫 cpw 4294   × cxp 5256  dom cdm 5258  *cxr 10257   < clt 10258  cle 10259  [,)cico 12362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-xr 10262  df-ico 12366
This theorem is referenced by:  ndmico  40288
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