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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 6881 | . . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘〈𝐴, 𝐵〉) | |
2 | ixx.1 | . . . . 5 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
3 | 2 | ixxf 12434 | . . . 4 ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
4 | 0elpw 5026 | . . . 4 ⊢ ∅ ∈ 𝒫 ℝ* | |
5 | 3, 4 | f0cli 6596 | . . 3 ⊢ (𝑂‘〈𝐴, 𝐵〉) ∈ 𝒫 ℝ* |
6 | 1, 5 | eqeltri 2874 | . 2 ⊢ (𝐴𝑂𝐵) ∈ 𝒫 ℝ* |
7 | ovex 6910 | . . 3 ⊢ (𝐴𝑂𝐵) ∈ V | |
8 | 7 | elpw 4355 | . 2 ⊢ ((𝐴𝑂𝐵) ∈ 𝒫 ℝ* ↔ (𝐴𝑂𝐵) ⊆ ℝ*) |
9 | 6, 8 | mpbi 222 | 1 ⊢ (𝐴𝑂𝐵) ⊆ ℝ* |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 385 = wceq 1653 ∈ wcel 2157 {crab 3093 ⊆ wss 3769 𝒫 cpw 4349 〈cop 4374 class class class wbr 4843 × cxp 5310 ‘cfv 6101 (class class class)co 6878 ↦ cmpt2 6880 ℝ*cxr 10362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-xr 10367 |
This theorem is referenced by: iccssxr 12505 iocssxr 12506 icossxr 12507 ioossioobi 40488 |
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