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Mirrors > Home > ILE Home > Th. List > unirnioo | GIF version |
Description: The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.) |
Ref | Expression |
---|---|
unirnioo | ⊢ ℝ = ∪ ran (,) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioomax 9699 | . . . 4 ⊢ (-∞(,)+∞) = ℝ | |
2 | ioof 9722 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
3 | ffn 5242 | . . . . . 6 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (,) Fn (ℝ* × ℝ*) |
5 | mnfxr 7790 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
6 | pnfxr 7786 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
7 | fnovrn 5886 | . . . . 5 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞(,)+∞) ∈ ran (,)) | |
8 | 4, 5, 6, 7 | mp3an 1300 | . . . 4 ⊢ (-∞(,)+∞) ∈ ran (,) |
9 | 1, 8 | eqeltrri 2191 | . . 3 ⊢ ℝ ∈ ran (,) |
10 | elssuni 3734 | . . 3 ⊢ (ℝ ∈ ran (,) → ℝ ⊆ ∪ ran (,)) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ ℝ ⊆ ∪ ran (,) |
12 | frn 5251 | . . . 4 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ) | |
13 | 2, 12 | ax-mp 5 | . . 3 ⊢ ran (,) ⊆ 𝒫 ℝ |
14 | sspwuni 3867 | . . 3 ⊢ (ran (,) ⊆ 𝒫 ℝ ↔ ∪ ran (,) ⊆ ℝ) | |
15 | 13, 14 | mpbi 144 | . 2 ⊢ ∪ ran (,) ⊆ ℝ |
16 | 11, 15 | eqssi 3083 | 1 ⊢ ℝ = ∪ ran (,) |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∈ wcel 1465 ⊆ wss 3041 𝒫 cpw 3480 ∪ cuni 3706 × cxp 4507 ran crn 4510 Fn wfn 5088 ⟶wf 5089 (class class class)co 5742 ℝcr 7587 +∞cpnf 7765 -∞cmnf 7766 ℝ*cxr 7767 (,)cioo 9639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-ioo 9643 |
This theorem is referenced by: uniretop 12621 tgioo 12642 |
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