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Mirrors > Home > ILE Home > Th. List > dfioo2 | GIF version |
Description: Alternate definition of the set of open intervals of extended reals. (Contributed by NM, 1-Mar-2007.) (Revised by Mario Carneiro, 1-Sep-2015.) |
Ref | Expression |
---|---|
dfioo2 | ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioof 9784 | . . 3 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
2 | ffn 5280 | . . 3 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
3 | fnovim 5887 | . . 3 ⊢ ((,) Fn (ℝ* × ℝ*) → (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ (𝑥(,)𝑦))) | |
4 | 1, 2, 3 | mp2b 8 | . 2 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ (𝑥(,)𝑦)) |
5 | iooval2 9728 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥(,)𝑦) = {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)}) | |
6 | 5 | mpoeq3ia 5844 | . 2 ⊢ (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ (𝑥(,)𝑦)) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)}) |
7 | 4, 6 | eqtri 2161 | 1 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)}) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 {crab 2421 𝒫 cpw 3515 class class class wbr 3937 × cxp 4545 Fn wfn 5126 ⟶wf 5127 (class class class)co 5782 ∈ cmpo 5784 ℝcr 7643 ℝ*cxr 7823 < clt 7824 (,)cioo 9701 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-ioo 9705 |
This theorem is referenced by: (None) |
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