unirnioo - Metamath Proof Explorer

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Theorem unirnioo 12886

Description: The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)

**Assertion**
Ref	Expression
unirnioo	⊢ ℝ = ∪ ran (,)

**Proof of Theorem unirnioo**
Step	Hyp	Ref	Expression
1		ioomax 12859	. . . 4 ⊢ (-∞(,)+∞) = ℝ
2		ioof 12884	. . . . . 6 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6502	. . . . . 6 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . . 5 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		mnfxr 10741	. . . . 5 ⊢ -∞ ∈ ℝ^*
6		pnfxr 10738	. . . . 5 ⊢ +∞ ∈ ℝ^*
7		fnovrn 7324	. . . . 5 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (-∞(,)+∞) ∈ ran (,))
8	4, 5, 6, 7	mp3an 1458	. . . 4 ⊢ (-∞(,)+∞) ∈ ran (,)
9	1, 8	eqeltrri 2849	. . 3 ⊢ ℝ ∈ ran (,)
10		elssuni 4833	. . 3 ⊢ (ℝ ∈ ran (,) → ℝ ⊆ ∪ ran (,))
11	9, 10	ax-mp 5	. 2 ⊢ ℝ ⊆ ∪ ran (,)
12		frn 6508	. . . 4 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ)
13	2, 12	ax-mp 5	. . 3 ⊢ ran (,) ⊆ 𝒫 ℝ
14		sspwuni 4990	. . 3 ⊢ (ran (,) ⊆ 𝒫 ℝ ↔ ∪ ran (,) ⊆ ℝ)
15	13, 14	mpbi 233	. 2 ⊢ ∪ ran (,) ⊆ ℝ
16	11, 15	eqssi 3910	1 ⊢ ℝ = ∪ ran (,)

Colors of variables: wff setvar class

Syntax hints: = wceq 1538 ∈ wcel 2111 ⊆ wss 3860 𝒫 cpw 4497 ∪ cuni 4801 × cxp 5525 ran crn 5528 Fn wfn 6334 ⟶wf 6335 (class class class)co 7155 ℝcr 10579 +∞cpnf 10715 -∞cmnf 10716 ℝ^*cxr 10717 (,)cioo 12784

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-cnex 10636 ax-resscn 10637 ax-pre-lttri 10654 ax-pre-lttrn 10655

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-iun 4888 df-br 5036 df-opab 5098 df-mpt 5116 df-id 5433 df-po 5446 df-so 5447 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7698 df-2nd 7699 df-er 8304 df-en 8533 df-dom 8534 df-sdom 8535 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-ioo 12788

This theorem is referenced by: pnfnei 21925 mnfnei 21926 uniretop 23469 tgioo 23502 xrtgioo 23512 bndth 23664 relowlssretop 35086 relowlpssretop 35087 mblfinlem3 35402 mblfinlem4 35403 ismblfin 35404

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