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Theorem icof 45229
Description: The set of left-closed right-open intervals of extended reals maps to subsets of extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Assertion
Ref Expression
icof [,):(ℝ* × ℝ*)⟶𝒫 ℝ*

Proof of Theorem icof
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2737 . . . 4 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} = {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
2 ssrab2 4079 . . . . 5 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ*
3 xrex 13030 . . . . . . 7 * ∈ V
43rabex 5338 . . . . . 6 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ V
54elpw 4603 . . . . 5 ({𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ*)
62, 5mpbir 231 . . . 4 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ*
71, 6eqeltrrdi 2849 . . 3 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ*)
87rgen2 3198 . 2 𝑥 ∈ ℝ*𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ*
9 df-ico 13394 . . 3 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
109fmpo 8094 . 2 (∀𝑥 ∈ ℝ*𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
118, 10mpbi 230 1 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2107  wral 3060  {crab 3435  wss 3950  𝒫 cpw 4599   class class class wbr 5142   × cxp 5682  wf 6556  *cxr 11295   < clt 11296  cle 11297  [,)cico 13390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-xr 11300  df-ico 13394
This theorem is referenced by:  fvvolicof  46011  volicoff  46015  voliooicof  46016  ovolval5lem2  46673  ovolval5lem3  46674  ovnovollem1  46676  ovnovollem2  46677
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