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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 12461 | . . . 4 ⊢ (-∞(,)+∞) = ℝ | |
2 | ioof 12484 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
3 | ffn 6206 | . . . . . 6 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (,) Fn (ℝ* × ℝ*) |
5 | mnfxr 10308 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
6 | pnfxr 10304 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
7 | fnovrn 6975 | . . . . 5 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞(,)+∞) ∈ ran (,)) | |
8 | 4, 5, 6, 7 | mp3an 1573 | . . . 4 ⊢ (-∞(,)+∞) ∈ ran (,) |
9 | 1, 8 | eqeltrri 2836 | . . 3 ⊢ ℝ ∈ ran (,) |
10 | elssuni 4619 | . . 3 ⊢ (ℝ ∈ ran (,) → ℝ ⊆ ∪ ran (,)) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ ℝ ⊆ ∪ ran (,) |
12 | frn 6214 | . . . 4 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ) | |
13 | 2, 12 | ax-mp 5 | . . 3 ⊢ ran (,) ⊆ 𝒫 ℝ |
14 | sspwuni 4763 | . . 3 ⊢ (ran (,) ⊆ 𝒫 ℝ ↔ ∪ ran (,) ⊆ ℝ) | |
15 | 13, 14 | mpbi 220 | . 2 ⊢ ∪ ran (,) ⊆ ℝ |
16 | 11, 15 | eqssi 3760 | 1 ⊢ ℝ = ∪ ran (,) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∈ wcel 2139 ⊆ wss 3715 𝒫 cpw 4302 ∪ cuni 4588 × cxp 5264 ran crn 5267 Fn wfn 6044 ⟶wf 6045 (class class class)co 6814 ℝcr 10147 +∞cpnf 10283 -∞cmnf 10284 ℝ*cxr 10285 (,)cioo 12388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-pre-lttri 10222 ax-pre-lttrn 10223 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-1st 7334 df-2nd 7335 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-ioo 12392 |
This theorem is referenced by: pnfnei 21246 mnfnei 21247 uniretop 22787 tgioo 22820 xrtgioo 22830 bndth 22978 relowlssretop 33540 relowlpssretop 33541 mblfinlem3 33779 mblfinlem4 33780 ismblfin 33781 |
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