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| Mirrors > Home > MPE Home > Th. List > Mathboxes > icof | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| icof | ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqidd 2737 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
| 2 | ssrab2 4079 | . . . . 5 ⊢ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ* | |
| 3 | xrex 13030 | . . . . . . 7 ⊢ ℝ* ∈ V | |
| 4 | 3 | rabex 5338 | . . . . . 6 ⊢ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ V | 
| 5 | 4 | elpw 4603 | . . . . 5 ⊢ ({𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ*) | 
| 6 | 2, 5 | mpbir 231 | . . . 4 ⊢ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ* | 
| 7 | 1, 6 | eqeltrrdi 2849 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ*) | 
| 8 | 7 | rgen2 3198 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ* | 
| 9 | df-ico 13394 | . . 3 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
| 10 | 9 | fmpo 8094 | . 2 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*) | 
| 11 | 8, 10 | mpbi 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 ℝ*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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