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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 7396 | . . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘〈𝐴, 𝐵〉) | |
2 | ixx.1 | . . . . 5 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
3 | 2 | ixxf 13316 | . . . 4 ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
4 | 0elpw 5347 | . . . 4 ⊢ ∅ ∈ 𝒫 ℝ* | |
5 | 3, 4 | f0cli 7084 | . . 3 ⊢ (𝑂‘〈𝐴, 𝐵〉) ∈ 𝒫 ℝ* |
6 | 1, 5 | eqeltri 2828 | . 2 ⊢ (𝐴𝑂𝐵) ∈ 𝒫 ℝ* |
7 | ovex 7426 | . . 3 ⊢ (𝐴𝑂𝐵) ∈ V | |
8 | 7 | elpw 4600 | . 2 ⊢ ((𝐴𝑂𝐵) ∈ 𝒫 ℝ* ↔ (𝐴𝑂𝐵) ⊆ ℝ*) |
9 | 6, 8 | mpbi 229 | 1 ⊢ (𝐴𝑂𝐵) ⊆ ℝ* |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3431 ⊆ wss 3944 𝒫 cpw 4596 〈cop 4628 class class class wbr 5141 × cxp 5667 ‘cfv 6532 (class class class)co 7393 ∈ cmpo 7395 ℝ*cxr 11229 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-xr 11234 |
This theorem is referenced by: iccssxr 13389 iocssxr 13390 icossxr 13391 ioossioobi 44001 |
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