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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 13399 | . . . 4 ⊢ (-∞(,)+∞) = ℝ | |
2 | ioof 13424 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
3 | ffn 6718 | . . . . . 6 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (,) Fn (ℝ* × ℝ*) |
5 | mnfxr 11271 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
6 | pnfxr 11268 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
7 | fnovrn 7582 | . . . . 5 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞(,)+∞) ∈ ran (,)) | |
8 | 4, 5, 6, 7 | mp3an 1462 | . . . 4 ⊢ (-∞(,)+∞) ∈ ran (,) |
9 | 1, 8 | eqeltrri 2831 | . . 3 ⊢ ℝ ∈ ran (,) |
10 | elssuni 4942 | . . 3 ⊢ (ℝ ∈ ran (,) → ℝ ⊆ ∪ ran (,)) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ ℝ ⊆ ∪ ran (,) |
12 | frn 6725 | . . . 4 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ) | |
13 | 2, 12 | ax-mp 5 | . . 3 ⊢ ran (,) ⊆ 𝒫 ℝ |
14 | sspwuni 5104 | . . 3 ⊢ (ran (,) ⊆ 𝒫 ℝ ↔ ∪ ran (,) ⊆ ℝ) | |
15 | 13, 14 | mpbi 229 | . 2 ⊢ ∪ ran (,) ⊆ ℝ |
16 | 11, 15 | eqssi 3999 | 1 ⊢ ℝ = ∪ ran (,) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ⊆ wss 3949 𝒫 cpw 4603 ∪ cuni 4909 × cxp 5675 ran crn 5678 Fn wfn 6539 ⟶wf 6540 (class class class)co 7409 ℝcr 11109 +∞cpnf 11245 -∞cmnf 11246 ℝ*cxr 11247 (,)cioo 13324 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-ioo 13328 |
This theorem is referenced by: pnfnei 22724 mnfnei 22725 uniretop 24279 tgioo 24312 xrtgioo 24322 bndth 24474 relowlssretop 36244 relowlpssretop 36245 mblfinlem3 36527 mblfinlem4 36528 ismblfin 36529 |
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