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Mirrors > Home > ILE Home > Th. List > ixxf | 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 | ssrab2 3019 | . . . 4 ⊢ {z ∈ ℝ* ∣ (x𝑅z ∧ z𝑆y)} ⊆ ℝ* | |
2 | xrex 8526 | . . . . 5 ⊢ ℝ* ∈ V | |
3 | 2 | elpw2 3902 | . . . 4 ⊢ ({z ∈ ℝ* ∣ (x𝑅z ∧ z𝑆y)} ∈ 𝒫 ℝ* ↔ {z ∈ ℝ* ∣ (x𝑅z ∧ z𝑆y)} ⊆ ℝ*) |
4 | 1, 3 | mpbir 134 | . . 3 ⊢ {z ∈ ℝ* ∣ (x𝑅z ∧ z𝑆y)} ∈ 𝒫 ℝ* |
5 | 4 | rgen2w 2371 | . 2 ⊢ ∀x ∈ ℝ* ∀y ∈ ℝ* {z ∈ ℝ* ∣ (x𝑅z ∧ z𝑆y)} ∈ 𝒫 ℝ* |
6 | ixx.1 | . . 3 ⊢ 𝑂 = (x ∈ ℝ*, y ∈ ℝ* ↦ {z ∈ ℝ* ∣ (x𝑅z ∧ z𝑆y)}) | |
7 | 6 | fmpt2 5769 | . 2 ⊢ (∀x ∈ ℝ* ∀y ∈ ℝ* {z ∈ ℝ* ∣ (x𝑅z ∧ z𝑆y)} ∈ 𝒫 ℝ* ↔ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*) |
8 | 5, 7 | mpbi 133 | 1 ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1242 ∈ wcel 1390 ∀wral 2300 {crab 2304 ⊆ wss 2911 𝒫 cpw 3351 class class class wbr 3755 × cxp 4286 ⟶wf 4841 ↦ cmpt2 5457 ℝ*cxr 6856 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-cnex 6774 ax-resscn 6775 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fv 4853 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-pnf 6859 df-mnf 6860 df-xr 6861 |
This theorem is referenced by: ixxex 8538 ixxssxr 8539 iccf 8611 |
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