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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 13179 | . . . 4 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
2 | ffn 6600 | . . . 4 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
3 | ovelrn 7448 | . . . 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 24739 | . . . . . . . 8 ⊢ ((𝑎(,)𝑏) ≠ ∅ → (𝐹‘(𝑎(,)𝑏)) = 〈𝑎, 𝑏〉) |
7 | 6 | fveq2d 6778 | . . . . . . 7 ⊢ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = ((,)‘〈𝑎, 𝑏〉)) |
8 | df-ov 7278 | . . . . . . 7 ⊢ (𝑎(,)𝑏) = ((,)‘〈𝑎, 𝑏〉) | |
9 | 7, 8 | eqtr4di 2796 | . . . . . 6 ⊢ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = (𝑎(,)𝑏)) |
10 | df-ne 2944 | . . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
11 | neeq1 3006 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → (𝐴 ≠ ∅ ↔ (𝑎(,)𝑏) ≠ ∅)) | |
12 | 10, 11 | bitr3id 285 | . . . . . . 7 ⊢ (𝐴 = (𝑎(,)𝑏) → (¬ 𝐴 = ∅ ↔ (𝑎(,)𝑏) ≠ ∅)) |
13 | 2fveq3 6779 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → ((,)‘(𝐹‘𝐴)) = ((,)‘(𝐹‘(𝑎(,)𝑏)))) | |
14 | id 22 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → 𝐴 = (𝑎(,)𝑏)) | |
15 | 13, 14 | eqeq12d 2754 | . . . . . . 7 ⊢ (𝐴 = (𝑎(,)𝑏) → (((,)‘(𝐹‘𝐴)) = 𝐴 ↔ ((,)‘(𝐹‘(𝑎(,)𝑏))) = (𝑎(,)𝑏))) |
16 | 12, 15 | imbi12d 345 | . . . . . 6 ⊢ (𝐴 = (𝑎(,)𝑏) → ((¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴) ↔ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = (𝑎(,)𝑏)))) |
17 | 9, 16 | mpbiri 257 | . . . . 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 3221 | . . 3 ⊢ (∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝐴 = (𝑎(,)𝑏) → (¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴)) |
20 | 4, 19 | sylbi 216 | . 2 ⊢ (𝐴 ∈ ran (,) → (¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴)) |
21 | ioorebas 13183 | . . . . . . 7 ⊢ (0(,)0) ∈ ran (,) | |
22 | 5 | ioorval 24738 | . . . . . . 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 13107 | . . . . . . 7 ⊢ (0(,)0) = ∅ | |
25 | 24 | iftruei 4466 | . . . . . 6 ⊢ if((0(,)0) = ∅, 〈0, 0〉, 〈inf((0(,)0), ℝ*, < ), sup((0(,)0), ℝ*, < )〉) = 〈0, 0〉 |
26 | 23, 25 | eqtri 2766 | . . . . 5 ⊢ (𝐹‘(0(,)0)) = 〈0, 0〉 |
27 | 26 | fveq2i 6777 | . . . 4 ⊢ ((,)‘(𝐹‘(0(,)0))) = ((,)‘〈0, 0〉) |
28 | df-ov 7278 | . . . 4 ⊢ (0(,)0) = ((,)‘〈0, 0〉) | |
29 | 27, 28 | eqtr4i 2769 | . . 3 ⊢ ((,)‘(𝐹‘(0(,)0))) = (0(,)0) |
30 | 24 | eqeq2i 2751 | . . . . . 6 ⊢ (𝐴 = (0(,)0) ↔ 𝐴 = ∅) |
31 | 30 | biimpri 227 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = (0(,)0)) |
32 | 31 | fveq2d 6778 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) = (𝐹‘(0(,)0))) |
33 | 32 | fveq2d 6778 | . . 3 ⊢ (𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = ((,)‘(𝐹‘(0(,)0)))) |
34 | 29, 33, 31 | 3eqtr4a 2804 | . 2 ⊢ (𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴) |
35 | 20, 34 | pm2.61d2 181 | 1 ⊢ (𝐴 ∈ ran (,) → ((,)‘(𝐹‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 ∅c0 4256 ifcif 4459 𝒫 cpw 4533 〈cop 4567 ↦ cmpt 5157 × cxp 5587 ran crn 5590 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 supcsup 9199 infcinf 9200 ℝcr 10870 0cc0 10871 ℝ*cxr 11008 < clt 11009 (,)cioo 13079 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-ioo 13083 |
This theorem is referenced by: uniioombllem2 24747 |
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