dfioo2 - Metamath Proof Explorer

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Theorem dfioo2 13431

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 13428	. . . 4 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		ffn 6717	. . . 4 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
3	1, 2	ax-mp 5	. . 3 ⊢ (,) Fn (ℝ^* × ℝ^*)
4		fnov 7542	. . 3 ⊢ ((,) Fn (ℝ^* × ℝ^) ↔ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^* ↦ (𝑥(,)𝑦)))
5	3, 4	mpbi 229	. 2 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ (𝑥(,)𝑦))
6		iooval2 13361	. . 3 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥(,)𝑦) = {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})
7	6	mpoeq3ia 7489	. 2 ⊢ (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ (𝑥(,)𝑦)) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})
8	5, 7	eqtri 2760	1 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})

Colors of variables: wff setvar class

Syntax hints: ∧ wa 396 = wceq 1541 {crab 3432 𝒫 cpw 4602 class class class wbr 5148 × cxp 5674 Fn wfn 6538 ⟶wf 6539 (class class class)co 7411 ∈ cmpo 7413 ℝcr 11111 ℝ^*cxr 11251 < clt 11252 (,)cioo 13328

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-ioo 13332

This theorem is referenced by: (None)

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