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Theorem icof 38885
 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 2622 . . . 4 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} = {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
2 ssrab2 3666 . . . . 5 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ*
3 xrex 11773 . . . . . . 7 * ∈ V
43rabex 4773 . . . . . 6 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ V
54elpw 4136 . . . . 5 ({𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ*)
62, 5mpbir 221 . . . 4 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ*
71, 6syl6eqelr 2707 . . 3 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ*)
87rgen2a 2971 . 2 𝑥 ∈ ℝ*𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ*
9 df-ico 12123 . . 3 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
109fmpt2 7182 . 2 (∀𝑥 ∈ ℝ*𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
118, 10mpbi 220 1 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 384   ∈ wcel 1987  ∀wral 2907  {crab 2911   ⊆ wss 3555  𝒫 cpw 4130   class class class wbr 4613   × cxp 5072  ⟶wf 5843  ℝ*cxr 10017   < clt 10018   ≤ cle 10019  [,)cico 12119 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-xr 10022  df-ico 12123 This theorem is referenced by:  fvvolicof  39515  volicoff  39519  voliooicof  39520  ovolval5lem2  40174  ovolval5lem3  40175  ovnovollem1  40177  ovnovollem2  40178
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