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Mirrors > Home > MPE Home > Th. List > ioof | Structured version Visualization version GIF version |
Description: The set of open intervals of extended reals maps to subsets of reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) |
Ref | Expression |
---|---|
ioof | ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iooval 12750 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥(,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | ioossre 12786 | . . . . 5 ⊢ (𝑥(,)𝑦) ⊆ ℝ | |
3 | ovex 7178 | . . . . . 6 ⊢ (𝑥(,)𝑦) ∈ V | |
4 | 3 | elpw 4542 | . . . . 5 ⊢ ((𝑥(,)𝑦) ∈ 𝒫 ℝ ↔ (𝑥(,)𝑦) ⊆ ℝ) |
5 | 2, 4 | mpbir 232 | . . . 4 ⊢ (𝑥(,)𝑦) ∈ 𝒫 ℝ |
6 | 1, 5 | syl6eqelr 2919 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ) |
7 | 6 | rgen2 3200 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ |
8 | df-ioo 12730 | . . 3 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
9 | 8 | fmpo 7755 | . 2 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ (,):(ℝ* × ℝ*)⟶𝒫 ℝ) |
10 | 7, 9 | mpbi 231 | 1 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∈ wcel 2105 ∀wral 3135 {crab 3139 ⊆ wss 3933 𝒫 cpw 4535 class class class wbr 5057 × cxp 5546 ⟶wf 6344 (class class class)co 7145 ℝcr 10524 ℝ*cxr 10662 < clt 10663 (,)cioo 12726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 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-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-ioo 12730 |
This theorem is referenced by: unirnioo 12825 dfioo2 12826 ioorebas 12827 qtopbaslem 23294 retopbas 23296 qdensere 23305 blssioo 23330 tgioo 23331 tgqioo 23335 re2ndc 23336 xrtgioo 23341 xrge0tsms 23369 bndth 23489 ovolfioo 23995 ovollb 24007 ovolicc2 24050 ovolfs2 24099 ioorf 24101 ioorinv 24104 ioorcl 24105 uniiccdif 24106 uniioovol 24107 uniiccvol 24108 uniioombllem2 24111 uniioombllem3a 24112 uniioombllem3 24113 uniioombllem4 24114 uniioombllem5 24115 uniioombl 24117 opnmblALT 24131 mbfdm 24154 mbfima 24158 mbfid 24163 ismbfd 24167 mbfimaopnlem 24183 i1fd 24209 xrge0tsmsd 30619 iccllysconn 32394 rellysconn 32395 relowlssretop 34526 relowlpssretop 34527 ftc1anc 34856 ftc2nc 34857 ioofun 41703 islptre 41776 volioof 42149 fvvolioof 42151 ovolval3 42806 ovolval4lem1 42808 ovolval5lem2 42812 ovolval5lem3 42813 |
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