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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 | elmpt2cl 5718 | . . 3 ⊢ (𝑥 ∈ (𝐴𝑂𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
3 | 1 | ixxf 8921 | . . . . . 6 ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
4 | 3 | fovcl 5626 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) ∈ 𝒫 ℝ*) |
5 | 4 | elpwid 3392 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) ⊆ ℝ*) |
6 | 5 | sseld 2998 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑥 ∈ (𝐴𝑂𝐵) → 𝑥 ∈ ℝ*)) |
7 | 2, 6 | mpcom 36 | . 2 ⊢ (𝑥 ∈ (𝐴𝑂𝐵) → 𝑥 ∈ ℝ*) |
8 | 7 | ssriv 3003 | 1 ⊢ (𝐴𝑂𝐵) ⊆ ℝ* |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1284 ∈ wcel 1433 {crab 2352 ⊆ wss 2973 𝒫 cpw 3382 class class class wbr 3785 (class class class)co 5532 ↦ cmpt2 5534 ℝ*cxr 7152 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-cnex 7067 ax-resscn 7068 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 |
This theorem is referenced by: iccssxr 8979 iocssxr 8980 icossxr 8981 |
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