![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 13350 | . . . 4 ⊢ (-∞(,)+∞) = ℝ | |
2 | ioof 13375 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
3 | ffn 6674 | . . . . . 6 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (,) Fn (ℝ* × ℝ*) |
5 | mnfxr 11222 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
6 | pnfxr 11219 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
7 | fnovrn 7535 | . . . . 5 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞(,)+∞) ∈ ran (,)) | |
8 | 4, 5, 6, 7 | mp3an 1462 | . . . 4 ⊢ (-∞(,)+∞) ∈ ran (,) |
9 | 1, 8 | eqeltrri 2830 | . . 3 ⊢ ℝ ∈ ran (,) |
10 | elssuni 4904 | . . 3 ⊢ (ℝ ∈ ran (,) → ℝ ⊆ ∪ ran (,)) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ ℝ ⊆ ∪ ran (,) |
12 | frn 6681 | . . . 4 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ) | |
13 | 2, 12 | ax-mp 5 | . . 3 ⊢ ran (,) ⊆ 𝒫 ℝ |
14 | sspwuni 5066 | . . 3 ⊢ (ran (,) ⊆ 𝒫 ℝ ↔ ∪ ran (,) ⊆ ℝ) | |
15 | 13, 14 | mpbi 229 | . 2 ⊢ ∪ ran (,) ⊆ ℝ |
16 | 11, 15 | eqssi 3964 | 1 ⊢ ℝ = ∪ ran (,) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ⊆ wss 3914 𝒫 cpw 4566 ∪ cuni 4871 × cxp 5637 ran crn 5640 Fn wfn 6497 ⟶wf 6498 (class class class)co 7363 ℝcr 11060 +∞cpnf 11196 -∞cmnf 11197 ℝ*cxr 11198 (,)cioo 13275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2703 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-cnex 11117 ax-resscn 11118 ax-pre-lttri 11135 ax-pre-lttrn 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4872 df-iun 4962 df-br 5112 df-opab 5174 df-mpt 5195 df-id 5537 df-po 5551 df-so 5552 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7927 df-2nd 7928 df-er 8656 df-en 8892 df-dom 8893 df-sdom 8894 df-pnf 11201 df-mnf 11202 df-xr 11203 df-ltxr 11204 df-le 11205 df-ioo 13279 |
This theorem is referenced by: pnfnei 22609 mnfnei 22610 uniretop 24164 tgioo 24197 xrtgioo 24207 bndth 24359 relowlssretop 35908 relowlpssretop 35909 mblfinlem3 36191 mblfinlem4 36192 ismblfin 36193 |
Copyright terms: Public domain | W3C validator |