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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 12070 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥(,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | ioossre 12106 | . . . . 5 ⊢ (𝑥(,)𝑦) ⊆ ℝ | |
3 | ovex 6577 | . . . . . 6 ⊢ (𝑥(,)𝑦) ∈ V | |
4 | 3 | elpw 4114 | . . . . 5 ⊢ ((𝑥(,)𝑦) ∈ 𝒫 ℝ ↔ (𝑥(,)𝑦) ⊆ ℝ) |
5 | 2, 4 | mpbir 220 | . . . 4 ⊢ (𝑥(,)𝑦) ∈ 𝒫 ℝ |
6 | 1, 5 | syl6eqelr 2697 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ) |
7 | 6 | rgen2a 2960 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ |
8 | df-ioo 12050 | . . 3 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
9 | 8 | fmpt2 7126 | . 2 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ (,):(ℝ* × ℝ*)⟶𝒫 ℝ) |
10 | 7, 9 | mpbi 219 | 1 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 1977 ∀wral 2896 {crab 2900 ⊆ wss 3540 𝒫 cpw 4108 class class class wbr 4583 × cxp 5036 ⟶wf 5800 (class class class)co 6549 ℝcr 9814 ℝ*cxr 9952 < clt 9953 (,)cioo 12046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ioo 12050 |
This theorem is referenced by: unirnioo 12144 dfioo2 12145 ioorebas 12146 qtopbaslem 22372 retopbas 22374 qdensere 22383 blssioo 22406 tgioo 22407 tgqioo 22411 re2ndc 22412 xrtgioo 22417 xrge0tsms 22445 bndth 22565 ovolfioo 23043 ovollb 23054 ovolicc2 23097 ovolfs2 23145 ioorf 23147 ioorinv 23150 ioorcl 23151 uniiccdif 23152 uniioovol 23153 uniiccvol 23154 uniioombllem2 23157 uniioombllem3a 23158 uniioombllem3 23159 uniioombllem4 23160 uniioombllem5 23161 uniioombl 23163 opnmblALT 23177 mbfdm 23201 mbfima 23205 mbfid 23209 ismbfd 23213 mbfimaopnlem 23228 i1fd 23254 xrge0tsmsd 29116 iccllyscon 30486 rellyscon 30487 relowlssretop 32387 relowlpssretop 32388 ftc1anc 32663 ftc2nc 32664 ioofun 38625 islptre 38686 volioof 38880 fvvolioof 38882 ovolval3 39537 ovolval4lem1 39539 ovolval5lem2 39543 ovolval5lem3 39544 |
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