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Mirrors > Home > MPE Home > Th. List > ixxssxr | Structured version Visualization version 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 |
---|---|
ixx.1 | ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) |
Ref | Expression |
---|---|
ixxssxr | ⊢ (𝐴𝑂𝐵) ⊆ ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7138 | . . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘〈𝐴, 𝐵〉) | |
2 | ixx.1 | . . . . 5 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
3 | 2 | ixxf 12736 | . . . 4 ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
4 | 0elpw 5221 | . . . 4 ⊢ ∅ ∈ 𝒫 ℝ* | |
5 | 3, 4 | f0cli 6841 | . . 3 ⊢ (𝑂‘〈𝐴, 𝐵〉) ∈ 𝒫 ℝ* |
6 | 1, 5 | eqeltri 2886 | . 2 ⊢ (𝐴𝑂𝐵) ∈ 𝒫 ℝ* |
7 | ovex 7168 | . . 3 ⊢ (𝐴𝑂𝐵) ∈ V | |
8 | 7 | elpw 4501 | . 2 ⊢ ((𝐴𝑂𝐵) ∈ 𝒫 ℝ* ↔ (𝐴𝑂𝐵) ⊆ ℝ*) |
9 | 6, 8 | mpbi 233 | 1 ⊢ (𝐴𝑂𝐵) ⊆ ℝ* |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 𝒫 cpw 4497 〈cop 4531 class class class wbr 5030 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ℝ*cxr 10663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-xr 10668 |
This theorem is referenced by: iccssxr 12808 iocssxr 12809 icossxr 12810 ioossioobi 42154 |
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