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Mirrors > Home > MPE Home > Th. List > ioorinv | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ioorf.1 | ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, 〈0, 0〉, 〈inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )〉)) |
Ref | Expression |
---|---|
ioorinv | ⊢ (𝐴 ∈ ran (,) → ((,)‘(𝐹‘𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioof 12825 | . . . 4 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
2 | ffn 6508 | . . . 4 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
3 | ovelrn 7313 | . . . 4 ⊢ ((,) Fn (ℝ* × ℝ*) → (𝐴 ∈ ran (,) ↔ ∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝐴 = (𝑎(,)𝑏))) | |
4 | 1, 2, 3 | mp2b 10 | . . 3 ⊢ (𝐴 ∈ ran (,) ↔ ∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝐴 = (𝑎(,)𝑏)) |
5 | ioorf.1 | . . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, 〈0, 0〉, 〈inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )〉)) | |
6 | 5 | ioorinv2 24105 | . . . . . . . 8 ⊢ ((𝑎(,)𝑏) ≠ ∅ → (𝐹‘(𝑎(,)𝑏)) = 〈𝑎, 𝑏〉) |
7 | 6 | fveq2d 6668 | . . . . . . 7 ⊢ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = ((,)‘〈𝑎, 𝑏〉)) |
8 | df-ov 7148 | . . . . . . 7 ⊢ (𝑎(,)𝑏) = ((,)‘〈𝑎, 𝑏〉) | |
9 | 7, 8 | syl6eqr 2874 | . . . . . 6 ⊢ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = (𝑎(,)𝑏)) |
10 | df-ne 3017 | . . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
11 | neeq1 3078 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → (𝐴 ≠ ∅ ↔ (𝑎(,)𝑏) ≠ ∅)) | |
12 | 10, 11 | syl5bbr 286 | . . . . . . 7 ⊢ (𝐴 = (𝑎(,)𝑏) → (¬ 𝐴 = ∅ ↔ (𝑎(,)𝑏) ≠ ∅)) |
13 | 2fveq3 6669 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → ((,)‘(𝐹‘𝐴)) = ((,)‘(𝐹‘(𝑎(,)𝑏)))) | |
14 | id 22 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → 𝐴 = (𝑎(,)𝑏)) | |
15 | 13, 14 | eqeq12d 2837 | . . . . . . 7 ⊢ (𝐴 = (𝑎(,)𝑏) → (((,)‘(𝐹‘𝐴)) = 𝐴 ↔ ((,)‘(𝐹‘(𝑎(,)𝑏))) = (𝑎(,)𝑏))) |
16 | 12, 15 | imbi12d 346 | . . . . . 6 ⊢ (𝐴 = (𝑎(,)𝑏) → ((¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴) ↔ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = (𝑎(,)𝑏)))) |
17 | 9, 16 | mpbiri 259 | . . . . 5 ⊢ (𝐴 = (𝑎(,)𝑏) → (¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴)) |
18 | 17 | a1i 11 | . . . 4 ⊢ ((𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) → (𝐴 = (𝑎(,)𝑏) → (¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴))) |
19 | 18 | rexlimivv 3292 | . . 3 ⊢ (∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝐴 = (𝑎(,)𝑏) → (¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴)) |
20 | 4, 19 | sylbi 218 | . 2 ⊢ (𝐴 ∈ ran (,) → (¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴)) |
21 | ioorebas 12829 | . . . . . . 7 ⊢ (0(,)0) ∈ ran (,) | |
22 | 5 | ioorval 24104 | . . . . . . 7 ⊢ ((0(,)0) ∈ ran (,) → (𝐹‘(0(,)0)) = if((0(,)0) = ∅, 〈0, 0〉, 〈inf((0(,)0), ℝ*, < ), sup((0(,)0), ℝ*, < )〉)) |
23 | 21, 22 | ax-mp 5 | . . . . . 6 ⊢ (𝐹‘(0(,)0)) = if((0(,)0) = ∅, 〈0, 0〉, 〈inf((0(,)0), ℝ*, < ), sup((0(,)0), ℝ*, < )〉) |
24 | iooid 12756 | . . . . . . 7 ⊢ (0(,)0) = ∅ | |
25 | 24 | iftruei 4472 | . . . . . 6 ⊢ if((0(,)0) = ∅, 〈0, 0〉, 〈inf((0(,)0), ℝ*, < ), sup((0(,)0), ℝ*, < )〉) = 〈0, 0〉 |
26 | 23, 25 | eqtri 2844 | . . . . 5 ⊢ (𝐹‘(0(,)0)) = 〈0, 0〉 |
27 | 26 | fveq2i 6667 | . . . 4 ⊢ ((,)‘(𝐹‘(0(,)0))) = ((,)‘〈0, 0〉) |
28 | df-ov 7148 | . . . 4 ⊢ (0(,)0) = ((,)‘〈0, 0〉) | |
29 | 27, 28 | eqtr4i 2847 | . . 3 ⊢ ((,)‘(𝐹‘(0(,)0))) = (0(,)0) |
30 | 24 | eqeq2i 2834 | . . . . . 6 ⊢ (𝐴 = (0(,)0) ↔ 𝐴 = ∅) |
31 | 30 | biimpri 229 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = (0(,)0)) |
32 | 31 | fveq2d 6668 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) = (𝐹‘(0(,)0))) |
33 | 32 | fveq2d 6668 | . . 3 ⊢ (𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = ((,)‘(𝐹‘(0(,)0)))) |
34 | 29, 33, 31 | 3eqtr4a 2882 | . 2 ⊢ (𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴) |
35 | 20, 34 | pm2.61d2 182 | 1 ⊢ (𝐴 ∈ ran (,) → ((,)‘(𝐹‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3016 ∃wrex 3139 ∅c0 4290 ifcif 4465 𝒫 cpw 4537 〈cop 4565 ↦ cmpt 5138 × cxp 5547 ran crn 5550 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7145 supcsup 8893 infcinf 8894 ℝcr 10525 0cc0 10526 ℝ*cxr 10663 < clt 10664 (,)cioo 12728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-sup 8895 df-inf 8896 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-n0 11887 df-z 11971 df-uz 12233 df-q 12338 df-ioo 12732 |
This theorem is referenced by: uniioombllem2 24113 |
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