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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 12838 | . . . 4 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
2 | ffn 6516 | . . . 4 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
3 | ovelrn 7326 | . . . 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 24178 | . . . . . . . 8 ⊢ ((𝑎(,)𝑏) ≠ ∅ → (𝐹‘(𝑎(,)𝑏)) = 〈𝑎, 𝑏〉) |
7 | 6 | fveq2d 6676 | . . . . . . 7 ⊢ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = ((,)‘〈𝑎, 𝑏〉)) |
8 | df-ov 7161 | . . . . . . 7 ⊢ (𝑎(,)𝑏) = ((,)‘〈𝑎, 𝑏〉) | |
9 | 7, 8 | syl6eqr 2876 | . . . . . 6 ⊢ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = (𝑎(,)𝑏)) |
10 | df-ne 3019 | . . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
11 | neeq1 3080 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → (𝐴 ≠ ∅ ↔ (𝑎(,)𝑏) ≠ ∅)) | |
12 | 10, 11 | syl5bbr 287 | . . . . . . 7 ⊢ (𝐴 = (𝑎(,)𝑏) → (¬ 𝐴 = ∅ ↔ (𝑎(,)𝑏) ≠ ∅)) |
13 | 2fveq3 6677 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → ((,)‘(𝐹‘𝐴)) = ((,)‘(𝐹‘(𝑎(,)𝑏)))) | |
14 | id 22 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → 𝐴 = (𝑎(,)𝑏)) | |
15 | 13, 14 | eqeq12d 2839 | . . . . . . 7 ⊢ (𝐴 = (𝑎(,)𝑏) → (((,)‘(𝐹‘𝐴)) = 𝐴 ↔ ((,)‘(𝐹‘(𝑎(,)𝑏))) = (𝑎(,)𝑏))) |
16 | 12, 15 | imbi12d 347 | . . . . . 6 ⊢ (𝐴 = (𝑎(,)𝑏) → ((¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴) ↔ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = (𝑎(,)𝑏)))) |
17 | 9, 16 | mpbiri 260 | . . . . 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 3294 | . . 3 ⊢ (∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝐴 = (𝑎(,)𝑏) → (¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴)) |
20 | 4, 19 | sylbi 219 | . 2 ⊢ (𝐴 ∈ ran (,) → (¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴)) |
21 | ioorebas 12842 | . . . . . . 7 ⊢ (0(,)0) ∈ ran (,) | |
22 | 5 | ioorval 24177 | . . . . . . 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 12769 | . . . . . . 7 ⊢ (0(,)0) = ∅ | |
25 | 24 | iftruei 4476 | . . . . . 6 ⊢ if((0(,)0) = ∅, 〈0, 0〉, 〈inf((0(,)0), ℝ*, < ), sup((0(,)0), ℝ*, < )〉) = 〈0, 0〉 |
26 | 23, 25 | eqtri 2846 | . . . . 5 ⊢ (𝐹‘(0(,)0)) = 〈0, 0〉 |
27 | 26 | fveq2i 6675 | . . . 4 ⊢ ((,)‘(𝐹‘(0(,)0))) = ((,)‘〈0, 0〉) |
28 | df-ov 7161 | . . . 4 ⊢ (0(,)0) = ((,)‘〈0, 0〉) | |
29 | 27, 28 | eqtr4i 2849 | . . 3 ⊢ ((,)‘(𝐹‘(0(,)0))) = (0(,)0) |
30 | 24 | eqeq2i 2836 | . . . . . 6 ⊢ (𝐴 = (0(,)0) ↔ 𝐴 = ∅) |
31 | 30 | biimpri 230 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = (0(,)0)) |
32 | 31 | fveq2d 6676 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) = (𝐹‘(0(,)0))) |
33 | 32 | fveq2d 6676 | . . 3 ⊢ (𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = ((,)‘(𝐹‘(0(,)0)))) |
34 | 29, 33, 31 | 3eqtr4a 2884 | . 2 ⊢ (𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴) |
35 | 20, 34 | pm2.61d2 183 | 1 ⊢ (𝐴 ∈ ran (,) → ((,)‘(𝐹‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 ∅c0 4293 ifcif 4469 𝒫 cpw 4541 〈cop 4575 ↦ cmpt 5148 × cxp 5555 ran crn 5558 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 supcsup 8906 infcinf 8907 ℝcr 10538 0cc0 10539 ℝ*cxr 10676 < clt 10677 (,)cioo 12741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-ioo 12745 |
This theorem is referenced by: uniioombllem2 24186 |
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