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Theorem ioorinv2 25552

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 13395	. . 3 ⊢ (𝐴(,)𝐵) ∈ ran (,)
2		ioorf.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩))
3	2	ioorval 25551	. . 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 4479	. . 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 4294	. . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴(,)𝐵))
7		eliooxr 13348	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
8	7	exlimiv 1932	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
9	6, 8	sylbi 217	. . . . . 6 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
10	9	simpld 494	. . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → 𝐴 ∈ ℝ^*)
11	9	simprd 495	. . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → 𝐵 ∈ ℝ^*)
12		id 22	. . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴(,)𝐵) ≠ ∅)
13		df-ioo 13293	. . . . . 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 13091	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵))
16		idd 24	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 < 𝑤))
17		xrltle 13091	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤))
18	13, 14, 15, 16, 17	ixxlb 13311	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴(,)𝐵) ≠ ∅) → inf((𝐴(,)𝐵), ℝ^*, < ) = 𝐴)
19	10, 11, 12, 18	syl3anc 1374	. . . 4 ⊢ ((𝐴(,)𝐵) ≠ ∅ → inf((𝐴(,)𝐵), ℝ^*, < ) = 𝐴)
20	13, 14, 15, 16, 17	ixxub 13310	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴(,)𝐵) ≠ ∅) → sup((𝐴(,)𝐵), ℝ^*, < ) = 𝐵)
21	10, 11, 12, 20	syl3anc 1374	. . . 4 ⊢ ((𝐴(,)𝐵) ≠ ∅ → sup((𝐴(,)𝐵), ℝ^*, < ) = 𝐵)
22	19, 21	opeq12d 4825	. . 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 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2933 ∅c0 4274 ifcif 4467 ⟨cop 4574 class class class wbr 5086 ↦ cmpt 5167 ran crn 5625 ‘cfv 6492 (class class class)co 7360 supcsup 9346 infcinf 9347 0cc0 11029 ℝ^*cxr 11169 < clt 11170 (,)cioo 13289

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9348 df-inf 9349 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-n0 12429 df-z 12516 df-uz 12780 df-q 12890 df-ioo 13293

This theorem is referenced by: ioorinv 25553 ioorcl 25554

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