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Mirrors > Home > ILE Home > Th. List > ioof | 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 9659 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥(,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | ioossre 9686 | . . . . 5 ⊢ (𝑥(,)𝑦) ⊆ ℝ | |
3 | df-ov 5745 | . . . . . . 7 ⊢ (𝑥(,)𝑦) = ((,)‘〈𝑥, 𝑦〉) | |
4 | iooex 9658 | . . . . . . . 8 ⊢ (,) ∈ V | |
5 | vex 2663 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
6 | vex 2663 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | opex 4121 | . . . . . . . 8 ⊢ 〈𝑥, 𝑦〉 ∈ V |
8 | 4, 7 | fvex 5409 | . . . . . . 7 ⊢ ((,)‘〈𝑥, 𝑦〉) ∈ V |
9 | 3, 8 | eqeltri 2190 | . . . . . 6 ⊢ (𝑥(,)𝑦) ∈ V |
10 | 9 | elpw 3486 | . . . . 5 ⊢ ((𝑥(,)𝑦) ∈ 𝒫 ℝ ↔ (𝑥(,)𝑦) ⊆ ℝ) |
11 | 2, 10 | mpbir 145 | . . . 4 ⊢ (𝑥(,)𝑦) ∈ 𝒫 ℝ |
12 | 1, 11 | syl6eqelr 2209 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ) |
13 | 12 | rgen2a 2463 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ |
14 | df-ioo 9643 | . . 3 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
15 | 14 | fmpo 6067 | . 2 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ (,):(ℝ* × ℝ*)⟶𝒫 ℝ) |
16 | 13, 15 | mpbi 144 | 1 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∈ wcel 1465 ∀wral 2393 {crab 2397 Vcvv 2660 ⊆ wss 3041 𝒫 cpw 3480 〈cop 3500 class class class wbr 3899 × cxp 4507 ⟶wf 5089 ‘cfv 5093 (class class class)co 5742 ℝcr 7587 ℝ*cxr 7767 < clt 7768 (,)cioo 9639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-ioo 9643 |
This theorem is referenced by: unirnioo 9724 dfioo2 9725 ioorebasg 9726 qtopbasss 12617 retopbas 12619 tgioo 12642 tgqioo 12643 |
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