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Theorem ioorinv2 25091

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 13427	. . 3 ⊢ (𝐴(,)𝐵) ∈ ran (,)
2		ioorf.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩))
3	2	ioorval 25090	. . 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 4540	. . 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 4346	. . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴(,)𝐵))
7		eliooxr 13381	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
8	7	exlimiv 1933	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
9	6, 8	sylbi 216	. . . . . 6 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
10	9	simpld 495	. . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → 𝐴 ∈ ℝ^*)
11	9	simprd 496	. . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → 𝐵 ∈ ℝ^*)
12		id 22	. . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴(,)𝐵) ≠ ∅)
13		df-ioo 13327	. . . . . 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 13127	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵))
16		idd 24	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 < 𝑤))
17		xrltle 13127	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤))
18	13, 14, 15, 16, 17	ixxlb 13345	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴(,)𝐵) ≠ ∅) → inf((𝐴(,)𝐵), ℝ^*, < ) = 𝐴)
19	10, 11, 12, 18	syl3anc 1371	. . . 4 ⊢ ((𝐴(,)𝐵) ≠ ∅ → inf((𝐴(,)𝐵), ℝ^*, < ) = 𝐴)
20	13, 14, 15, 16, 17	ixxub 13344	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴(,)𝐵) ≠ ∅) → sup((𝐴(,)𝐵), ℝ^*, < ) = 𝐵)
21	10, 11, 12, 20	syl3anc 1371	. . . 4 ⊢ ((𝐴(,)𝐵) ≠ ∅ → sup((𝐴(,)𝐵), ℝ^*, < ) = 𝐵)
22	19, 21	opeq12d 4881	. . 3 ⊢ ((𝐴(,)𝐵) ≠ ∅ → ⟨inf((𝐴(,)𝐵), ℝ^, < ), sup((𝐴(,)𝐵), ℝ^, < )⟩ = ⟨𝐴, 𝐵⟩)
23	5, 22	eqtrd 2772	. 2 ⊢ ((𝐴(,)𝐵) ≠ ∅ → if((𝐴(,)𝐵) = ∅, ⟨0, 0⟩, ⟨inf((𝐴(,)𝐵), ℝ^, < ), sup((𝐴(,)𝐵), ℝ^, < )⟩) = ⟨𝐴, 𝐵⟩)
24	4, 23	eqtrid 2784	1 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐹‘(𝐴(,)𝐵)) = ⟨𝐴, 𝐵⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∅c0 4322 ifcif 4528 ⟨cop 4634 class class class wbr 5148 ↦ cmpt 5231 ran crn 5677 ‘cfv 6543 (class class class)co 7408 supcsup 9434 infcinf 9435 0cc0 11109 ℝ^*cxr 11246 < clt 11247 (,)cioo 13323

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 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 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-rmo 3376 df-reu 3377 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-pss 3967 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-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 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-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-ioo 13327

This theorem is referenced by: ioorinv 25092 ioorcl 25093

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