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Theorem icof 41466
 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 2820 . . . 4 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} = {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
2 ssrab2 4054 . . . . 5 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ*
3 xrex 12378 . . . . . . 7 * ∈ V
43rabex 5226 . . . . . 6 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ V
54elpw 4544 . . . . 5 ({𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ*)
62, 5mpbir 233 . . . 4 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ*
71, 6eqeltrrdi 2920 . . 3 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ*)
87rgen2 3201 . 2 𝑥 ∈ ℝ*𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ*
9 df-ico 12736 . . 3 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
109fmpo 7758 . 2 (∀𝑥 ∈ ℝ*𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
118, 10mpbi 232 1 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 398   ∈ wcel 2107  ∀wral 3136  {crab 3140   ⊆ wss 3934  𝒫 cpw 4537   class class class wbr 5057   × cxp 5546  ⟶wf 6344  ℝ*cxr 10666   < clt 10667   ≤ cle 10668  [,)cico 12732 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-xr 10671  df-ico 12736 This theorem is referenced by:  fvvolicof  42261  volicoff  42265  voliooicof  42266  ovolval5lem2  42920  ovolval5lem3  42921  ovnovollem1  42923  ovnovollem2  42924
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