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Mirrors > Home > MPE Home > Th. List > ioof | Structured version Visualization version GIF version |
Description: The set of open intervals of extended reals maps to subsets of reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) |
Ref | Expression |
---|---|
ioof | ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iooval 13372 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥(,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | ioossre 13409 | . . . . 5 ⊢ (𝑥(,)𝑦) ⊆ ℝ | |
3 | ovex 7447 | . . . . . 6 ⊢ (𝑥(,)𝑦) ∈ V | |
4 | 3 | elpw 4602 | . . . . 5 ⊢ ((𝑥(,)𝑦) ∈ 𝒫 ℝ ↔ (𝑥(,)𝑦) ⊆ ℝ) |
5 | 2, 4 | mpbir 230 | . . . 4 ⊢ (𝑥(,)𝑦) ∈ 𝒫 ℝ |
6 | 1, 5 | eqeltrrdi 2837 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ) |
7 | 6 | rgen2 3192 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ |
8 | df-ioo 13352 | . . 3 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
9 | 8 | fmpo 8066 | . 2 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ (,):(ℝ* × ℝ*)⟶𝒫 ℝ) |
10 | 7, 9 | mpbi 229 | 1 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∈ wcel 2099 ∀wral 3056 {crab 3427 ⊆ wss 3944 𝒫 cpw 4598 class class class wbr 5142 × cxp 5670 ⟶wf 6538 (class class class)co 7414 ℝcr 11129 ℝ*cxr 11269 < clt 11270 (,)cioo 13348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-pre-lttri 11204 ax-pre-lttrn 11205 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-ioo 13352 |
This theorem is referenced by: unirnioo 13450 dfioo2 13451 ioorebas 13452 qtopbaslem 24662 retopbas 24664 qdensere 24673 blssioo 24698 tgioo 24699 tgqioo 24703 re2ndc 24704 xrtgioo 24709 xrge0tsms 24737 bndth 24871 ovolfioo 25383 ovollb 25395 ovolicc2 25438 ovolfs2 25487 ioorf 25489 ioorinv 25492 ioorcl 25493 uniiccdif 25494 uniioovol 25495 uniiccvol 25496 uniioombllem2 25499 uniioombllem3a 25500 uniioombllem3 25501 uniioombllem4 25502 uniioombllem5 25503 uniioombl 25505 opnmblALT 25519 mbfdm 25542 mbfima 25546 mbfid 25551 ismbfd 25555 mbfimaopnlem 25571 i1fd 25597 xrge0tsmsd 32749 iccllysconn 34796 rellysconn 34797 relowlssretop 36778 relowlpssretop 36779 ftc1anc 37109 ftc2nc 37110 ioofun 44859 islptre 44930 volioof 45298 fvvolioof 45300 ovolval3 45958 ovolval4lem1 45960 ovolval5lem2 45964 ovolval5lem3 45965 |
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