| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 7434 | . . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘〈𝐴, 𝐵〉) | |
| 2 | ixx.1 | . . . . 5 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
| 3 | 2 | ixxf 13397 | . . . 4 ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
| 4 | 0elpw 5356 | . . . 4 ⊢ ∅ ∈ 𝒫 ℝ* | |
| 5 | 3, 4 | f0cli 7118 | . . 3 ⊢ (𝑂‘〈𝐴, 𝐵〉) ∈ 𝒫 ℝ* |
| 6 | 1, 5 | eqeltri 2837 | . 2 ⊢ (𝐴𝑂𝐵) ∈ 𝒫 ℝ* |
| 7 | ovex 7464 | . . 3 ⊢ (𝐴𝑂𝐵) ∈ V | |
| 8 | 7 | elpw 4604 | . 2 ⊢ ((𝐴𝑂𝐵) ∈ 𝒫 ℝ* ↔ (𝐴𝑂𝐵) ⊆ ℝ*) |
| 9 | 6, 8 | mpbi 230 | 1 ⊢ (𝐴𝑂𝐵) ⊆ ℝ* |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ⊆ wss 3951 𝒫 cpw 4600 〈cop 4632 class class class wbr 5143 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℝ*cxr 11294 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-xr 11299 |
| This theorem is referenced by: iccssxr 13470 iocssxr 13471 icossxr 13472 ioossioobi 45530 |
| Copyright terms: Public domain | W3C validator |