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Mirrors > Home > ILE Home > Th. List > ixxssxr | GIF version |
Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.) |
Ref | Expression |
---|---|
ixxssxr.1 | ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) |
Ref | Expression |
---|---|
ixxssxr | ⊢ (𝐴𝑂𝐵) ⊆ ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixxssxr.1 | . . . 4 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
2 | 1 | elmpocl 5920 | . . 3 ⊢ (𝑥 ∈ (𝐴𝑂𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
3 | 1 | ixxf 9568 | . . . . . 6 ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
4 | 3 | fovcl 5828 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) ∈ 𝒫 ℝ*) |
5 | 4 | elpwid 3485 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) ⊆ ℝ*) |
6 | 5 | sseld 3060 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑥 ∈ (𝐴𝑂𝐵) → 𝑥 ∈ ℝ*)) |
7 | 2, 6 | mpcom 36 | . 2 ⊢ (𝑥 ∈ (𝐴𝑂𝐵) → 𝑥 ∈ ℝ*) |
8 | 7 | ssriv 3065 | 1 ⊢ (𝐴𝑂𝐵) ⊆ ℝ* |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1312 ∈ wcel 1461 {crab 2392 ⊆ wss 3035 𝒫 cpw 3474 class class class wbr 3893 (class class class)co 5726 ∈ cmpo 5728 ℝ*cxr 7717 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-cnex 7630 ax-resscn 7631 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fv 5087 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-pnf 7720 df-mnf 7721 df-xr 7722 |
This theorem is referenced by: iccssxr 9626 iocssxr 9627 icossxr 9628 |
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