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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 9907 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥(,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | ioossre 9934 | . . . . 5 ⊢ (𝑥(,)𝑦) ⊆ ℝ | |
3 | df-ov 5877 | . . . . . . 7 ⊢ (𝑥(,)𝑦) = ((,)‘⟨𝑥, 𝑦⟩) | |
4 | iooex 9906 | . . . . . . . 8 ⊢ (,) ∈ V | |
5 | vex 2740 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
6 | vex 2740 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | opex 4229 | . . . . . . . 8 ⊢ ⟨𝑥, 𝑦⟩ ∈ V |
8 | 4, 7 | fvex 5535 | . . . . . . 7 ⊢ ((,)‘⟨𝑥, 𝑦⟩) ∈ V |
9 | 3, 8 | eqeltri 2250 | . . . . . 6 ⊢ (𝑥(,)𝑦) ∈ V |
10 | 9 | elpw 3581 | . . . . 5 ⊢ ((𝑥(,)𝑦) ∈ 𝒫 ℝ ↔ (𝑥(,)𝑦) ⊆ ℝ) |
11 | 2, 10 | mpbir 146 | . . . 4 ⊢ (𝑥(,)𝑦) ∈ 𝒫 ℝ |
12 | 1, 11 | eqeltrrdi 2269 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ) |
13 | 12 | rgen2a 2531 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ |
14 | df-ioo 9891 | . . 3 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
15 | 14 | fmpo 6201 | . 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 2148 ∀wral 2455 {crab 2459 Vcvv 2737 ⊆ wss 3129 𝒫 cpw 3575 ⟨cop 3595 class class class wbr 4003 × cxp 4624 ⟶wf 5212 ‘cfv 5216 (class class class)co 5874 ℝcr 7809 ℝ*cxr 7990 < clt 7991 (,)cioo 9887 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-po 4296 df-iso 4297 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-ioo 9891 |
This theorem is referenced by: unirnioo 9972 dfioo2 9973 ioorebasg 9974 qtopbasss 13991 retopbas 13993 tgioo 14016 tgqioo 14017 |
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