dfioo2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dfioo2		Structured version Visualization version GIF version

Theorem dfioo2 13467

Description: Alternate definition of the set of open intervals of extended reals. (Contributed by NM, 1-Mar-2007.) (Revised by Mario Carneiro, 1-Sep-2015.)

**Assertion**
Ref	Expression
dfioo2	⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})

Distinct variable group: 𝑥,𝑤,𝑦

**Proof of Theorem dfioo2**
Step	Hyp	Ref	Expression
1		ioof 13464	. . . 4 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		ffn 6706	. . . 4 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
3	1, 2	ax-mp 5	. . 3 ⊢ (,) Fn (ℝ^* × ℝ^*)
4		fnov 7538	. . 3 ⊢ ((,) Fn (ℝ^* × ℝ^) ↔ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^* ↦ (𝑥(,)𝑦)))
5	3, 4	mpbi 230	. 2 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ (𝑥(,)𝑦))
6		iooval2 13395	. . 3 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥(,)𝑦) = {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})
7	6	mpoeq3ia 7485	. 2 ⊢ (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ (𝑥(,)𝑦)) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})
8	5, 7	eqtri 2758	1 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 {crab 3415 𝒫 cpw 4575 class class class wbr 5119 × cxp 5652 Fn wfn 6526 ⟶wf 6527 (class class class)co 7405 ∈ cmpo 7407 ℝcr 11128 ℝ^*cxr 11268 < clt 11269 (,)cioo 13362

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-pre-lttri 11203 ax-pre-lttrn 11204

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7988 df-2nd 7989 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-ioo 13366

This theorem is referenced by: (None)

W3C validator