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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 9974 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥(,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | ioossre 10001 | . . . . 5 ⊢ (𝑥(,)𝑦) ⊆ ℝ | |
3 | df-ov 5921 | . . . . . . 7 ⊢ (𝑥(,)𝑦) = ((,)‘〈𝑥, 𝑦〉) | |
4 | iooex 9973 | . . . . . . . 8 ⊢ (,) ∈ V | |
5 | vex 2763 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
6 | vex 2763 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | opex 4258 | . . . . . . . 8 ⊢ 〈𝑥, 𝑦〉 ∈ V |
8 | 4, 7 | fvex 5574 | . . . . . . 7 ⊢ ((,)‘〈𝑥, 𝑦〉) ∈ V |
9 | 3, 8 | eqeltri 2266 | . . . . . 6 ⊢ (𝑥(,)𝑦) ∈ V |
10 | 9 | elpw 3607 | . . . . 5 ⊢ ((𝑥(,)𝑦) ∈ 𝒫 ℝ ↔ (𝑥(,)𝑦) ⊆ ℝ) |
11 | 2, 10 | mpbir 146 | . . . 4 ⊢ (𝑥(,)𝑦) ∈ 𝒫 ℝ |
12 | 1, 11 | eqeltrrdi 2285 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ) |
13 | 12 | rgen2a 2548 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ |
14 | df-ioo 9958 | . . 3 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
15 | 14 | fmpo 6254 | . 2 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ (,):(ℝ* × ℝ*)⟶𝒫 ℝ) |
16 | 13, 15 | mpbi 145 | 1 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∈ wcel 2164 ∀wral 2472 {crab 2476 Vcvv 2760 ⊆ wss 3153 𝒫 cpw 3601 〈cop 3621 class class class wbr 4029 × cxp 4657 ⟶wf 5250 ‘cfv 5254 (class class class)co 5918 ℝcr 7871 ℝ*cxr 8053 < clt 8054 (,)cioo 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-po 4327 df-iso 4328 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-ioo 9958 |
This theorem is referenced by: unirnioo 10039 dfioo2 10040 ioorebasg 10041 qtopbasss 14689 retopbas 14691 tgioo 14714 tgqioo 14715 |
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