dfioo2 - Metamath Proof Explorer

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Theorem dfioo2 13487

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 13484	. . . 4 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		ffn 6737	. . . 4 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
3	1, 2	ax-mp 5	. . 3 ⊢ (,) Fn (ℝ^* × ℝ^*)
4		fnov 7564	. . 3 ⊢ ((,) Fn (ℝ^* × ℝ^) ↔ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^* ↦ (𝑥(,)𝑦)))
5	3, 4	mpbi 230	. 2 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ (𝑥(,)𝑦))
6		iooval2 13417	. . 3 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥(,)𝑦) = {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})
7	6	mpoeq3ia 7511	. 2 ⊢ (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ (𝑥(,)𝑦)) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})
8	5, 7	eqtri 2763	1 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1537 {crab 3433 𝒫 cpw 4605 class class class wbr 5148 × cxp 5687 Fn wfn 6558 ⟶wf 6559 (class class class)co 7431 ∈ cmpo 7433 ℝcr 11152 ℝ^*cxr 11292 < clt 11293 (,)cioo 13384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ioo 13388

This theorem is referenced by: (None)

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