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Theorem ioorf 25090

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 13424	. . . 4 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6718	. . . 4 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4		ovelrn 7583	. . . 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 12313	. . . . . . . . 9 ⊢ 0 ≤ 0
7		df-br 5150	. . . . . . . . 9 ⊢ (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
8	6, 7	mpbi 229	. . . . . . . 8 ⊢ ⟨0, 0⟩ ∈ ≤
9		0xr 11261	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
10		opelxpi 5714	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 0 ∈ ℝ^) → ⟨0, 0⟩ ∈ (ℝ^ × ℝ^*))
11	9, 9, 10	mp2an 691	. . . . . . . 8 ⊢ ⟨0, 0⟩ ∈ (ℝ^* × ℝ^*)
12	8, 11	elini 4194	. . . . . . 7 ⊢ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ^* × ℝ^*))
13	12	a1i 11	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ 𝑥 = ∅) → ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ^ × ℝ^*)))
14		simplr 768	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → 𝑥 = (𝑎(,)𝑏))
15	14	infeq1d 9472	. . . . . . . . 9 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → inf(𝑥, ℝ^, < ) = inf((𝑎(,)𝑏), ℝ^*, < ))
16		simplll 774	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → 𝑎 ∈ ℝ^)
17		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → 𝑏 ∈ ℝ^)
18		simpr 486	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ¬ 𝑥 = ∅)
19	18	neqned 2948	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → 𝑥 ≠ ∅)
20	14, 19	eqnetrrd 3010	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → (𝑎(,)𝑏) ≠ ∅)
21		df-ioo 13328	. . . . . . . . . . 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 13128	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) → (𝑤 < 𝑏 → 𝑤 ≤ 𝑏))
24		idd 24	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝑎 < 𝑤 → 𝑎 < 𝑤))
25		xrltle 13128	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝑎 < 𝑤 → 𝑎 ≤ 𝑤))
26	21, 22, 23, 24, 25	ixxlb 13346	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^* ∧ (𝑎(,)𝑏) ≠ ∅) → inf((𝑎(,)𝑏), ℝ^*, < ) = 𝑎)
27	16, 17, 20, 26	syl3anc 1372	. . . . . . . . 9 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → inf((𝑎(,)𝑏), ℝ^, < ) = 𝑎)
28	15, 27	eqtrd 2773	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → inf(𝑥, ℝ^, < ) = 𝑎)
29	14	supeq1d 9441	. . . . . . . . 9 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → sup(𝑥, ℝ^, < ) = sup((𝑎(,)𝑏), ℝ^*, < ))
30	21, 22, 23, 24, 25	ixxub 13345	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^* ∧ (𝑎(,)𝑏) ≠ ∅) → sup((𝑎(,)𝑏), ℝ^*, < ) = 𝑏)
31	16, 17, 20, 30	syl3anc 1372	. . . . . . . . 9 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → sup((𝑎(,)𝑏), ℝ^, < ) = 𝑏)
32	29, 31	eqtrd 2773	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → sup(𝑥, ℝ^, < ) = 𝑏)
33	28, 32	opeq12d 4882	. . . . . . 7 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^*, < )⟩ = ⟨𝑎, 𝑏⟩)
34		ioon0 13350	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) → ((𝑎(,)𝑏) ≠ ∅ ↔ 𝑎 < 𝑏))
35	34	ad2antrr 725	. . . . . . . . . . 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 13128	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) → (𝑎 < 𝑏 → 𝑎 ≤ 𝑏))
38	37	ad2antrr 725	. . . . . . . . . 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 5150	. . . . . . . . 9 ⊢ (𝑎 ≤ 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ ≤ )
41	39, 40	sylib 217	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ⟨𝑎, 𝑏⟩ ∈ ≤ )
42		opelxpi 5714	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) → ⟨𝑎, 𝑏⟩ ∈ (ℝ^ × ℝ^*))
43	42	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ⟨𝑎, 𝑏⟩ ∈ (ℝ^ × ℝ^*))
44	41, 43	elind 4195	. . . . . . 7 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ⟨𝑎, 𝑏⟩ ∈ ( ≤ ∩ (ℝ^ × ℝ^*)))
45	33, 44	eqeltrd 2834	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) ∧ ¬ 𝑥 = ∅) → ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩ ∈ ( ≤ ∩ (ℝ^ × ℝ^*)))
46	13, 45	ifclda 4564	. . . . 5 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑥 = (𝑎(,)𝑏)) → if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩) ∈ ( ≤ ∩ (ℝ^ × ℝ^*)))
47	46	ex 414	. . . 4 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) → (𝑥 = (𝑎(,)𝑏) → if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩) ∈ ( ≤ ∩ (ℝ^ × ℝ^*))))
48	47	rexlimivv 3200	. . 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 7112	1 ⊢ 𝐹:ran (,)⟶( ≤ ∩ (ℝ^* × ℝ^*))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 ∩ cin 3948 ∅c0 4323 ifcif 4529 𝒫 cpw 4603 ⟨cop 4635 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 ran crn 5678 Fn wfn 6539 ⟶wf 6540 (class class class)co 7409 supcsup 9435 infcinf 9436 ℝcr 11109 0cc0 11110 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 (,)cioo 13324

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-ioo 13328

This theorem is referenced by: ioorcl 25094 uniioombllem2 25100

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