dfioo2 - Metamath Proof Explorer

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Theorem dfioo2 13418

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 13415	. . . 4 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		ffn 6691	. . . 4 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
3	1, 2	ax-mp 5	. . 3 ⊢ (,) Fn (ℝ^* × ℝ^*)
4		fnov 7523	. . 3 ⊢ ((,) Fn (ℝ^* × ℝ^) ↔ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^* ↦ (𝑥(,)𝑦)))
5	3, 4	mpbi 230	. 2 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ (𝑥(,)𝑦))
6		iooval2 13346	. . 3 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥(,)𝑦) = {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})
7	6	mpoeq3ia 7470	. 2 ⊢ (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ (𝑥(,)𝑦)) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})
8	5, 7	eqtri 2753	1 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 {crab 3408 𝒫 cpw 4566 class class class wbr 5110 × cxp 5639 Fn wfn 6509 ⟶wf 6510 (class class class)co 7390 ∈ cmpo 7392 ℝcr 11074 ℝ^*cxr 11214 < clt 11215 (,)cioo 13313

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 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-pre-lttri 11149 ax-pre-lttrn 11150

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 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-ioo 13317

This theorem is referenced by: (None)

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