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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 13411 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥(,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
| 2 | ioossre 13448 | . . . . 5 ⊢ (𝑥(,)𝑦) ⊆ ℝ | |
| 3 | ovex 7464 | . . . . . 6 ⊢ (𝑥(,)𝑦) ∈ V | |
| 4 | 3 | elpw 4604 | . . . . 5 ⊢ ((𝑥(,)𝑦) ∈ 𝒫 ℝ ↔ (𝑥(,)𝑦) ⊆ ℝ) |
| 5 | 2, 4 | mpbir 231 | . . . 4 ⊢ (𝑥(,)𝑦) ∈ 𝒫 ℝ |
| 6 | 1, 5 | eqeltrrdi 2850 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ) |
| 7 | 6 | rgen2 3199 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ |
| 8 | df-ioo 13391 | . . 3 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
| 9 | 8 | fmpo 8093 | . 2 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ (,):(ℝ* × ℝ*)⟶𝒫 ℝ) |
| 10 | 7, 9 | mpbi 230 | 1 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 𝒫 cpw 4600 class class class wbr 5143 × cxp 5683 ⟶wf 6557 (class class class)co 7431 ℝcr 11154 ℝ*cxr 11294 < clt 11295 (,)cioo 13387 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 |
| This theorem is referenced by: unirnioo 13489 dfioo2 13490 ioorebas 13491 qtopbaslem 24779 retopbas 24781 qdensere 24790 blssioo 24816 tgioo 24817 tgqioo 24821 re2ndc 24822 xrtgioo 24828 xrge0tsms 24856 bndth 24990 ovolfioo 25502 ovollb 25514 ovolicc2 25557 ovolfs2 25606 ioorf 25608 ioorinv 25611 ioorcl 25612 uniiccdif 25613 uniioovol 25614 uniiccvol 25615 uniioombllem2 25618 uniioombllem3a 25619 uniioombllem3 25620 uniioombllem4 25621 uniioombllem5 25622 uniioombl 25624 opnmblALT 25638 mbfdm 25661 mbfima 25665 mbfid 25670 ismbfd 25674 mbfimaopnlem 25690 i1fd 25716 xrge0tsmsd 33065 iccllysconn 35255 rellysconn 35256 relowlssretop 37364 relowlpssretop 37365 ftc1anc 37708 ftc2nc 37709 ioofun 45564 islptre 45634 volioof 46002 fvvolioof 46004 ovolval3 46662 ovolval4lem1 46664 ovolval5lem2 46668 ovolval5lem3 46669 |
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