ioorinv2 - Metamath Proof Explorer

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Theorem ioorinv2 23389

Description: The function 𝐹 is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)

**Hypothesis**
Ref	Expression
ioorf.1	⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩))

**Assertion**
Ref	Expression
ioorinv2	⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐹‘(𝐴(,)𝐵)) = ⟨𝐴, 𝐵⟩)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 Allowed substitution hint: 𝐹(𝑥)

Proof of Theorem ioorinv2 Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioorebas 12313	. . 3 ⊢ (𝐴(,)𝐵) ∈ ran (,)
2		ioorf.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩))
3	2	ioorval 23388	. . 3 ⊢ ((𝐴(,)𝐵) ∈ ran (,) → (𝐹‘(𝐴(,)𝐵)) = if((𝐴(,)𝐵) = ∅, ⟨0, 0⟩, ⟨inf((𝐴(,)𝐵), ℝ^, < ), sup((𝐴(,)𝐵), ℝ^, < )⟩))
4	1, 3	ax-mp 5	. 2 ⊢ (𝐹‘(𝐴(,)𝐵)) = if((𝐴(,)𝐵) = ∅, ⟨0, 0⟩, ⟨inf((𝐴(,)𝐵), ℝ^, < ), sup((𝐴(,)𝐵), ℝ^, < )⟩)
5		ifnefalse 4131	. . 3 ⊢ ((𝐴(,)𝐵) ≠ ∅ → if((𝐴(,)𝐵) = ∅, ⟨0, 0⟩, ⟨inf((𝐴(,)𝐵), ℝ^, < ), sup((𝐴(,)𝐵), ℝ^, < )⟩) = ⟨inf((𝐴(,)𝐵), ℝ^, < ), sup((𝐴(,)𝐵), ℝ^, < )⟩)
6		n0 3964	. . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴(,)𝐵))
7		eliooxr 12270	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
8	7	exlimiv 1898	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
9	6, 8	sylbi 207	. . . . . 6 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
10	9	simpld 474	. . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → 𝐴 ∈ ℝ^*)
11	9	simprd 478	. . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → 𝐵 ∈ ℝ^*)
12		id 22	. . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴(,)𝐵) ≠ ∅)
13		df-ioo 12217	. . . . . 6 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
14		idd 24	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤 < 𝐵))
15		xrltle 12020	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵))
16		idd 24	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 < 𝑤))
17		xrltle 12020	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤))
18	13, 14, 15, 16, 17	ixxlb 12235	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴(,)𝐵) ≠ ∅) → inf((𝐴(,)𝐵), ℝ^*, < ) = 𝐴)
19	10, 11, 12, 18	syl3anc 1366	. . . 4 ⊢ ((𝐴(,)𝐵) ≠ ∅ → inf((𝐴(,)𝐵), ℝ^*, < ) = 𝐴)
20	13, 14, 15, 16, 17	ixxub 12234	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴(,)𝐵) ≠ ∅) → sup((𝐴(,)𝐵), ℝ^*, < ) = 𝐵)
21	10, 11, 12, 20	syl3anc 1366	. . . 4 ⊢ ((𝐴(,)𝐵) ≠ ∅ → sup((𝐴(,)𝐵), ℝ^*, < ) = 𝐵)
22	19, 21	opeq12d 4441	. . 3 ⊢ ((𝐴(,)𝐵) ≠ ∅ → ⟨inf((𝐴(,)𝐵), ℝ^, < ), sup((𝐴(,)𝐵), ℝ^, < )⟩ = ⟨𝐴, 𝐵⟩)
23	5, 22	eqtrd 2685	. 2 ⊢ ((𝐴(,)𝐵) ≠ ∅ → if((𝐴(,)𝐵) = ∅, ⟨0, 0⟩, ⟨inf((𝐴(,)𝐵), ℝ^, < ), sup((𝐴(,)𝐵), ℝ^, < )⟩) = ⟨𝐴, 𝐵⟩)
24	4, 23	syl5eq 2697	1 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐹‘(𝐴(,)𝐵)) = ⟨𝐴, 𝐵⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∃wex 1744 ∈ wcel 2030 ≠ wne 2823 ∅c0 3948 ifcif 4119 ⟨cop 4216 class class class wbr 4685 ↦ cmpt 4762 ran crn 5144 ‘cfv 5926 (class class class)co 6690 supcsup 8387 infcinf 8388 0cc0 9974 ℝ^*cxr 10111 < clt 10112 (,)cioo 12213

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-inf 8390 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-q 11827 df-ioo 12217

This theorem is referenced by: ioorinv 23390 ioorcl 23391

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