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| Mirrors > Home > MPE Home > Th. List > unirnioo | Structured version Visualization version 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 13463 | . . . 4 ⊢ (-∞(,)+∞) = ℝ | |
| 2 | ioof 13488 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
| 3 | ffn 6735 | . . . . . 6 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
| 4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (,) Fn (ℝ* × ℝ*) | 
| 5 | mnfxr 11319 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
| 6 | pnfxr 11316 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
| 7 | fnovrn 7609 | . . . . 5 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞(,)+∞) ∈ ran (,)) | |
| 8 | 4, 5, 6, 7 | mp3an 1462 | . . . 4 ⊢ (-∞(,)+∞) ∈ ran (,) | 
| 9 | 1, 8 | eqeltrri 2837 | . . 3 ⊢ ℝ ∈ ran (,) | 
| 10 | elssuni 4936 | . . 3 ⊢ (ℝ ∈ ran (,) → ℝ ⊆ ∪ ran (,)) | |
| 11 | 9, 10 | ax-mp 5 | . 2 ⊢ ℝ ⊆ ∪ ran (,) | 
| 12 | frn 6742 | . . . 4 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ) | |
| 13 | 2, 12 | ax-mp 5 | . . 3 ⊢ ran (,) ⊆ 𝒫 ℝ | 
| 14 | sspwuni 5099 | . . 3 ⊢ (ran (,) ⊆ 𝒫 ℝ ↔ ∪ ran (,) ⊆ ℝ) | |
| 15 | 13, 14 | mpbi 230 | . 2 ⊢ ∪ ran (,) ⊆ ℝ | 
| 16 | 11, 15 | eqssi 3999 | 1 ⊢ ℝ = ∪ ran (,) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 𝒫 cpw 4599 ∪ cuni 4906 × cxp 5682 ran crn 5685 Fn wfn 6555 ⟶wf 6556 (class class class)co 7432 ℝcr 11155 +∞cpnf 11293 -∞cmnf 11294 ℝ*cxr 11295 (,)cioo 13388 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 ax-pre-lttrn 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-ioo 13392 | 
| This theorem is referenced by: pnfnei 23229 mnfnei 23230 uniretop 24784 tgioo 24818 xrtgioo 24829 bndth 24991 relowlssretop 37365 relowlpssretop 37366 mblfinlem3 37667 mblfinlem4 37668 ismblfin 37669 | 
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