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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 8817 | . . . 4 ⊢ (-∞(,)+∞) = ℝ | |
2 | ioof 8840 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
3 | ffn 5046 | . . . . . 6 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
4 | 2, 3 | ax-mp 7 | . . . . 5 ⊢ (,) Fn (ℝ* × ℝ*) |
5 | mnfxr 8694 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
6 | pnfxr 8692 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
7 | fnovrn 5648 | . . . . 5 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞(,)+∞) ∈ ran (,)) | |
8 | 4, 5, 6, 7 | mp3an 1232 | . . . 4 ⊢ (-∞(,)+∞) ∈ ran (,) |
9 | 1, 8 | eqeltrri 2111 | . . 3 ⊢ ℝ ∈ ran (,) |
10 | elssuni 3608 | . . 3 ⊢ (ℝ ∈ ran (,) → ℝ ⊆ ∪ ran (,)) | |
11 | 9, 10 | ax-mp 7 | . 2 ⊢ ℝ ⊆ ∪ ran (,) |
12 | frn 5052 | . . . 4 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ) | |
13 | 2, 12 | ax-mp 7 | . . 3 ⊢ ran (,) ⊆ 𝒫 ℝ |
14 | sspwuni 3739 | . . 3 ⊢ (ran (,) ⊆ 𝒫 ℝ ↔ ∪ ran (,) ⊆ ℝ) | |
15 | 13, 14 | mpbi 133 | . 2 ⊢ ∪ ran (,) ⊆ ℝ |
16 | 11, 15 | eqssi 2961 | 1 ⊢ ℝ = ∪ ran (,) |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 ⊆ wss 2917 𝒫 cpw 3359 ∪ cuni 3580 × cxp 4343 ran crn 4346 Fn wfn 4897 ⟶wf 4898 (class class class)co 5512 ℝcr 6888 +∞cpnf 7057 -∞cmnf 7058 ℝ*cxr 7059 (,)cioo 8757 |
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-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-po 4033 df-iso 4034 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-ioo 8761 |
This theorem is referenced by: (None) |
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