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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 9865 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥(,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | ioossre 9892 | . . . . 5 ⊢ (𝑥(,)𝑦) ⊆ ℝ | |
3 | df-ov 5856 | . . . . . . 7 ⊢ (𝑥(,)𝑦) = ((,)‘〈𝑥, 𝑦〉) | |
4 | iooex 9864 | . . . . . . . 8 ⊢ (,) ∈ V | |
5 | vex 2733 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
6 | vex 2733 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | opex 4214 | . . . . . . . 8 ⊢ 〈𝑥, 𝑦〉 ∈ V |
8 | 4, 7 | fvex 5516 | . . . . . . 7 ⊢ ((,)‘〈𝑥, 𝑦〉) ∈ V |
9 | 3, 8 | eqeltri 2243 | . . . . . 6 ⊢ (𝑥(,)𝑦) ∈ V |
10 | 9 | elpw 3572 | . . . . 5 ⊢ ((𝑥(,)𝑦) ∈ 𝒫 ℝ ↔ (𝑥(,)𝑦) ⊆ ℝ) |
11 | 2, 10 | mpbir 145 | . . . 4 ⊢ (𝑥(,)𝑦) ∈ 𝒫 ℝ |
12 | 1, 11 | eqeltrrdi 2262 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ) |
13 | 12 | rgen2a 2524 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ |
14 | df-ioo 9849 | . . 3 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
15 | 14 | fmpo 6180 | . 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 2141 ∀wral 2448 {crab 2452 Vcvv 2730 ⊆ wss 3121 𝒫 cpw 3566 〈cop 3586 class class class wbr 3989 × cxp 4609 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 ℝcr 7773 ℝ*cxr 7953 < clt 7954 (,)cioo 9845 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-ioo 9849 |
This theorem is referenced by: unirnioo 9930 dfioo2 9931 ioorebasg 9932 qtopbasss 13315 retopbas 13317 tgioo 13340 tgqioo 13341 |
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