unirnioo - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > unirnioo		Structured version Visualization version GIF version

Theorem unirnioo 13416

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 13389	. . . 4 ⊢ (-∞(,)+∞) = ℝ
2		ioof 13414	. . . . . 6 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6690	. . . . . 6 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . . 5 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		mnfxr 11237	. . . . 5 ⊢ -∞ ∈ ℝ^*
6		pnfxr 11234	. . . . 5 ⊢ +∞ ∈ ℝ^*
7		fnovrn 7566	. . . . 5 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (-∞(,)+∞) ∈ ran (,))
8	4, 5, 6, 7	mp3an 1463	. . . 4 ⊢ (-∞(,)+∞) ∈ ran (,)
9	1, 8	eqeltrri 2826	. . 3 ⊢ ℝ ∈ ran (,)
10		elssuni 4903	. . 3 ⊢ (ℝ ∈ ran (,) → ℝ ⊆ ∪ ran (,))
11	9, 10	ax-mp 5	. 2 ⊢ ℝ ⊆ ∪ ran (,)
12		frn 6697	. . . 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 230	. 2 ⊢ ∪ ran (,) ⊆ ℝ
16	11, 15	eqssi 3965	1 ⊢ ℝ = ∪ ran (,)

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 ⊆ wss 3916 𝒫 cpw 4565 ∪ cuni 4873 × cxp 5638 ran crn 5641 Fn wfn 6508 ⟶wf 6509 (class class class)co 7389 ℝcr 11073 +∞cpnf 11211 -∞cmnf 11212 ℝ^*cxr 11213 (,)cioo 13312

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-pre-lttri 11148 ax-pre-lttrn 11149

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-po 5548 df-so 5549 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-ov 7392 df-oprab 7393 df-mpo 7394 df-1st 7970 df-2nd 7971 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-ioo 13316

This theorem is referenced by: pnfnei 23113 mnfnei 23114 uniretop 24656 tgioo 24690 xrtgioo 24701 bndth 24863 relowlssretop 37346 relowlpssretop 37347 mblfinlem3 37648 mblfinlem4 37649 ismblfin 37650

W3C validator