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Mirrors > Home > MPE Home > Th. List > ixxf | Structured version Visualization version GIF version |
Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) |
Ref | Expression |
---|---|
ixx.1 | ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) |
Ref | Expression |
---|---|
ixxf | ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrex 12715 | . . . 4 ⊢ ℝ* ∈ V | |
2 | ssrab2 4013 | . . . 4 ⊢ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} ⊆ ℝ* | |
3 | 1, 2 | elpwi2 5269 | . . 3 ⊢ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} ∈ 𝒫 ℝ* |
4 | 3 | rgen2w 3077 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} ∈ 𝒫 ℝ* |
5 | ixx.1 | . . 3 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
6 | 5 | fmpo 7898 | . 2 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)} ∈ 𝒫 ℝ* ↔ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*) |
7 | 4, 6 | mpbi 229 | 1 ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 Vcvv 3430 𝒫 cpw 4534 class class class wbr 5074 × cxp 5583 ⟶wf 6423 ∈ cmpo 7270 ℝ*cxr 10996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5222 ax-nul 5229 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-fv 6435 df-oprab 7272 df-mpo 7273 df-1st 7821 df-2nd 7822 df-xr 11001 |
This theorem is referenced by: ixxex 13078 ixxssxr 13079 elixx3g 13080 ndmioo 13094 iccf 13168 iocpnfordt 22354 icomnfordt 22355 tpr2rico 31848 icoreresf 35509 icoreelrn 35518 relowlpssretop 35521 dmico 43062 |
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