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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 13381 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥(,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | ioossre 13418 | . . . . 5 ⊢ (𝑥(,)𝑦) ⊆ ℝ | |
3 | ovex 7453 | . . . . . 6 ⊢ (𝑥(,)𝑦) ∈ V | |
4 | 3 | elpw 4607 | . . . . 5 ⊢ ((𝑥(,)𝑦) ∈ 𝒫 ℝ ↔ (𝑥(,)𝑦) ⊆ ℝ) |
5 | 2, 4 | mpbir 230 | . . . 4 ⊢ (𝑥(,)𝑦) ∈ 𝒫 ℝ |
6 | 1, 5 | eqeltrrdi 2838 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ) |
7 | 6 | rgen2 3194 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ |
8 | df-ioo 13361 | . . 3 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
9 | 8 | fmpo 8072 | . 2 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ (,):(ℝ* × ℝ*)⟶𝒫 ℝ) |
10 | 7, 9 | mpbi 229 | 1 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∈ wcel 2099 ∀wral 3058 {crab 3429 ⊆ wss 3947 𝒫 cpw 4603 class class class wbr 5148 × cxp 5676 ⟶wf 6544 (class class class)co 7420 ℝcr 11138 ℝ*cxr 11278 < clt 11279 (,)cioo 13357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-pre-lttri 11213 ax-pre-lttrn 11214 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-ioo 13361 |
This theorem is referenced by: unirnioo 13459 dfioo2 13460 ioorebas 13461 qtopbaslem 24688 retopbas 24690 qdensere 24699 blssioo 24724 tgioo 24725 tgqioo 24729 re2ndc 24730 xrtgioo 24735 xrge0tsms 24763 bndth 24897 ovolfioo 25409 ovollb 25421 ovolicc2 25464 ovolfs2 25513 ioorf 25515 ioorinv 25518 ioorcl 25519 uniiccdif 25520 uniioovol 25521 uniiccvol 25522 uniioombllem2 25525 uniioombllem3a 25526 uniioombllem3 25527 uniioombllem4 25528 uniioombllem5 25529 uniioombl 25531 opnmblALT 25545 mbfdm 25568 mbfima 25572 mbfid 25577 ismbfd 25581 mbfimaopnlem 25597 i1fd 25623 xrge0tsmsd 32784 iccllysconn 34860 rellysconn 34861 relowlssretop 36842 relowlpssretop 36843 ftc1anc 37174 ftc2nc 37175 ioofun 44936 islptre 45007 volioof 45375 fvvolioof 45377 ovolval3 46035 ovolval4lem1 46037 ovolval5lem2 46041 ovolval5lem3 46042 |
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