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Theorem icof 45162
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 2736 . . . 4 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} = {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
2 ssrab2 4090 . . . . 5 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ*
3 xrex 13027 . . . . . . 7 * ∈ V
43rabex 5345 . . . . . 6 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ V
54elpw 4609 . . . . 5 ({𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ*)
62, 5mpbir 231 . . . 4 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ*
71, 6eqeltrrdi 2848 . . 3 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ*)
87rgen2 3197 . 2 𝑥 ∈ ℝ*𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ*
9 df-ico 13390 . . 3 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
109fmpo 8092 . 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 2106  wral 3059  {crab 3433  wss 3963  𝒫 cpw 4605   class class class wbr 5148   × cxp 5687  wf 6559  *cxr 11292   < clt 11293  cle 11294  [,)cico 13386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-xr 11297  df-ico 13390
This theorem is referenced by:  fvvolicof  45947  volicoff  45951  voliooicof  45952  ovolval5lem2  46609  ovolval5lem3  46610  ovnovollem1  46612  ovnovollem2  46613
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