Mathbox for Glauco Siliprandi |
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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 2623 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | ssrab2 3687 | . . . . 5 ⊢ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ* | |
3 | xrex 11829 | . . . . . . 7 ⊢ ℝ* ∈ V | |
4 | 3 | rabex 4813 | . . . . . 6 ⊢ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ V |
5 | 4 | elpw 4164 | . . . . 5 ⊢ ({𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ*) |
6 | 2, 5 | mpbir 221 | . . . 4 ⊢ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ* |
7 | 1, 6 | syl6eqelr 2710 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ*) |
8 | 7 | rgen2a 2977 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ* |
9 | df-ico 12181 | . . 3 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
10 | 9 | fmpt2 7237 | . 2 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*) |
11 | 8, 10 | mpbi 220 | 1 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∈ wcel 1990 ∀wral 2912 {crab 2916 ⊆ wss 3574 𝒫 cpw 4158 class class class wbr 4653 × cxp 5112 ⟶wf 5884 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 [,)cico 12177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-xr 10078 df-ico 12181 |
This theorem is referenced by: fvvolicof 40208 volicoff 40212 voliooicof 40213 ovolval5lem2 40867 ovolval5lem3 40868 ovnovollem1 40870 ovnovollem2 40871 |
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