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Theorem ioorf 25089

Description: Define a function from open intervals to their endpoints. (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
ioorf	⊢ 𝐹:ran (,)⟶( ≤ ∩ (ℝ^* × ℝ^*))

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

Step	Hyp	Ref	Expression
1		ioorf.1	. 2 ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩))
2		ioof 13423	. . . 4 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6717	. . . 4 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4		ovelrn 7582	. . . 4 ⊢ ((,) Fn (ℝ^* × ℝ^) → (𝑥 ∈ ran (,) ↔ ∃𝑎 ∈ ℝ^ ∃𝑏 ∈ ℝ^* 𝑥 = (𝑎(,)𝑏)))
5	2, 3, 4	mp2b 10	. . 3 ⊢ (𝑥 ∈ ran (,) ↔ ∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* 𝑥 = (𝑎(,)𝑏))
6		0le0 12312	. . . . . . . . 9 ⊢ 0 ≤ 0
7		df-br 5149	. . . . . . . . 9 ⊢ (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
8	6, 7	mpbi 229	. . . . . . . 8 ⊢ ⟨0, 0⟩ ∈ ≤
9		0xr 11260	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
10		opelxpi 5713	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 0 ∈ ℝ^) → ⟨0, 0⟩ ∈ (ℝ^ × ℝ^*))
11	9, 9, 10	mp2an 690	. . . . . . . 8 ⊢ ⟨0, 0⟩ ∈ (ℝ^* × ℝ^*)
12	8, 11	elini 4193	. . . . . . 7 ⊢ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ^* × ℝ^*))
13	12	a1i 11	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ 𝑥 = ∅) → ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ^ × ℝ^*)))
14		simplr 767	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → 𝑥 = (𝑎(,)𝑏))
15	14	infeq1d 9471	. . . . . . . . 9 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → inf(𝑥, ℝ^, < ) = inf((𝑎(,)𝑏), ℝ^*, < ))
16		simplll 773	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → 𝑎 ∈ ℝ^)
17		simpllr 774	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → 𝑏 ∈ ℝ^)
18		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ¬ 𝑥 = ∅)
19	18	neqned 2947	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → 𝑥 ≠ ∅)
20	14, 19	eqnetrrd 3009	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → (𝑎(,)𝑏) ≠ ∅)
21		df-ioo 13327	. . . . . . . . . . 11 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
22		idd 24	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) → (𝑤 < 𝑏 → 𝑤 < 𝑏))
23		xrltle 13127	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) → (𝑤 < 𝑏 → 𝑤 ≤ 𝑏))
24		idd 24	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝑎 < 𝑤 → 𝑎 < 𝑤))
25		xrltle 13127	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝑎 < 𝑤 → 𝑎 ≤ 𝑤))
26	21, 22, 23, 24, 25	ixxlb 13345	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^* ∧ (𝑎(,)𝑏) ≠ ∅) → inf((𝑎(,)𝑏), ℝ^*, < ) = 𝑎)
27	16, 17, 20, 26	syl3anc 1371	. . . . . . . . 9 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → inf((𝑎(,)𝑏), ℝ^, < ) = 𝑎)
28	15, 27	eqtrd 2772	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → inf(𝑥, ℝ^, < ) = 𝑎)
29	14	supeq1d 9440	. . . . . . . . 9 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → sup(𝑥, ℝ^, < ) = sup((𝑎(,)𝑏), ℝ^*, < ))
30	21, 22, 23, 24, 25	ixxub 13344	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^* ∧ (𝑎(,)𝑏) ≠ ∅) → sup((𝑎(,)𝑏), ℝ^*, < ) = 𝑏)
31	16, 17, 20, 30	syl3anc 1371	. . . . . . . . 9 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → sup((𝑎(,)𝑏), ℝ^, < ) = 𝑏)
32	29, 31	eqtrd 2772	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → sup(𝑥, ℝ^, < ) = 𝑏)
33	28, 32	opeq12d 4881	. . . . . . 7 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^*, < )⟩ = ⟨𝑎, 𝑏⟩)
34		ioon0 13349	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) → ((𝑎(,)𝑏) ≠ ∅ ↔ 𝑎 < 𝑏))
35	34	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ((𝑎(,)𝑏) ≠ ∅ ↔ 𝑎 < 𝑏))
36	20, 35	mpbid 231	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → 𝑎 < 𝑏)
37		xrltle 13127	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) → (𝑎 < 𝑏 → 𝑎 ≤ 𝑏))
38	37	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → (𝑎 < 𝑏 → 𝑎 ≤ 𝑏))
39	36, 38	mpd 15	. . . . . . . . 9 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → 𝑎 ≤ 𝑏)
40		df-br 5149	. . . . . . . . 9 ⊢ (𝑎 ≤ 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ ≤ )
41	39, 40	sylib 217	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ⟨𝑎, 𝑏⟩ ∈ ≤ )
42		opelxpi 5713	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) → ⟨𝑎, 𝑏⟩ ∈ (ℝ^ × ℝ^*))
43	42	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ⟨𝑎, 𝑏⟩ ∈ (ℝ^ × ℝ^*))
44	41, 43	elind 4194	. . . . . . 7 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ⟨𝑎, 𝑏⟩ ∈ ( ≤ ∩ (ℝ^ × ℝ^*)))
45	33, 44	eqeltrd 2833	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩ ∈ ( ≤ ∩ (ℝ^ × ℝ^*)))
46	13, 45	ifclda 4563	. . . . 5 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) → if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩) ∈ ( ≤ ∩ (ℝ^ × ℝ^*)))
47	46	ex 413	. . . 4 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) → (𝑥 = (𝑎(,)𝑏) → if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩) ∈ ( ≤ ∩ (ℝ^ × ℝ^*))))
48	47	rexlimivv 3199	. . 3 ⊢ (∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* 𝑥 = (𝑎(,)𝑏) → if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩) ∈ ( ≤ ∩ (ℝ^* × ℝ^*)))
49	5, 48	sylbi 216	. 2 ⊢ (𝑥 ∈ ran (,) → if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩) ∈ ( ≤ ∩ (ℝ^* × ℝ^*)))
50	1, 49	fmpti 7111	1 ⊢ 𝐹:ran (,)⟶( ≤ ∩ (ℝ^* × ℝ^*))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 ∩ cin 3947 ∅c0 4322 ifcif 4528 𝒫 cpw 4602 ⟨cop 4634 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ran crn 5677 Fn wfn 6538 ⟶wf 6539 (class class class)co 7408 supcsup 9434 infcinf 9435 ℝcr 11108 0cc0 11109 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 (,)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: ioorcl 25093 uniioombllem2 25099

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