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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 9684 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥(,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | ioossre 9711 | . . . . 5 ⊢ (𝑥(,)𝑦) ⊆ ℝ | |
3 | df-ov 5770 | . . . . . . 7 ⊢ (𝑥(,)𝑦) = ((,)‘〈𝑥, 𝑦〉) | |
4 | iooex 9683 | . . . . . . . 8 ⊢ (,) ∈ V | |
5 | vex 2684 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
6 | vex 2684 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | opex 4146 | . . . . . . . 8 ⊢ 〈𝑥, 𝑦〉 ∈ V |
8 | 4, 7 | fvex 5434 | . . . . . . 7 ⊢ ((,)‘〈𝑥, 𝑦〉) ∈ V |
9 | 3, 8 | eqeltri 2210 | . . . . . 6 ⊢ (𝑥(,)𝑦) ∈ V |
10 | 9 | elpw 3511 | . . . . 5 ⊢ ((𝑥(,)𝑦) ∈ 𝒫 ℝ ↔ (𝑥(,)𝑦) ⊆ ℝ) |
11 | 2, 10 | mpbir 145 | . . . 4 ⊢ (𝑥(,)𝑦) ∈ 𝒫 ℝ |
12 | 1, 11 | eqeltrrdi 2229 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ) |
13 | 12 | rgen2a 2484 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ |
14 | df-ioo 9668 | . . 3 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
15 | 14 | fmpo 6092 | . 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 1480 ∀wral 2414 {crab 2418 Vcvv 2681 ⊆ wss 3066 𝒫 cpw 3505 〈cop 3525 class class class wbr 3924 × cxp 4532 ⟶wf 5114 ‘cfv 5118 (class class class)co 5767 ℝcr 7612 ℝ*cxr 7792 < clt 7793 (,)cioo 9664 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-ioo 9668 |
This theorem is referenced by: unirnioo 9749 dfioo2 9750 ioorebasg 9751 qtopbasss 12679 retopbas 12681 tgioo 12704 tgqioo 12705 |
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