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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 13487 | . . . 4 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
| 2 | ffn 6736 | . . . 4 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
| 3 | ovelrn 7609 | . . . 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 25610 | . . . . . . . 8 ⊢ ((𝑎(,)𝑏) ≠ ∅ → (𝐹‘(𝑎(,)𝑏)) = 〈𝑎, 𝑏〉) | 
| 7 | 6 | fveq2d 6910 | . . . . . . 7 ⊢ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = ((,)‘〈𝑎, 𝑏〉)) | 
| 8 | df-ov 7434 | . . . . . . 7 ⊢ (𝑎(,)𝑏) = ((,)‘〈𝑎, 𝑏〉) | |
| 9 | 7, 8 | eqtr4di 2795 | . . . . . 6 ⊢ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = (𝑎(,)𝑏)) | 
| 10 | df-ne 2941 | . . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
| 11 | neeq1 3003 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → (𝐴 ≠ ∅ ↔ (𝑎(,)𝑏) ≠ ∅)) | |
| 12 | 10, 11 | bitr3id 285 | . . . . . . 7 ⊢ (𝐴 = (𝑎(,)𝑏) → (¬ 𝐴 = ∅ ↔ (𝑎(,)𝑏) ≠ ∅)) | 
| 13 | 2fveq3 6911 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → ((,)‘(𝐹‘𝐴)) = ((,)‘(𝐹‘(𝑎(,)𝑏)))) | |
| 14 | id 22 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → 𝐴 = (𝑎(,)𝑏)) | |
| 15 | 13, 14 | eqeq12d 2753 | . . . . . . 7 ⊢ (𝐴 = (𝑎(,)𝑏) → (((,)‘(𝐹‘𝐴)) = 𝐴 ↔ ((,)‘(𝐹‘(𝑎(,)𝑏))) = (𝑎(,)𝑏))) | 
| 16 | 12, 15 | imbi12d 344 | . . . . . 6 ⊢ (𝐴 = (𝑎(,)𝑏) → ((¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴) ↔ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = (𝑎(,)𝑏)))) | 
| 17 | 9, 16 | mpbiri 258 | . . . . 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 3201 | . . 3 ⊢ (∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝐴 = (𝑎(,)𝑏) → (¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴)) | 
| 20 | 4, 19 | sylbi 217 | . 2 ⊢ (𝐴 ∈ ran (,) → (¬ 𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = 𝐴)) | 
| 21 | ioorebas 13491 | . . . . . . 7 ⊢ (0(,)0) ∈ ran (,) | |
| 22 | 5 | ioorval 25609 | . . . . . . 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 13415 | . . . . . . 7 ⊢ (0(,)0) = ∅ | |
| 25 | 24 | iftruei 4532 | . . . . . 6 ⊢ if((0(,)0) = ∅, 〈0, 0〉, 〈inf((0(,)0), ℝ*, < ), sup((0(,)0), ℝ*, < )〉) = 〈0, 0〉 | 
| 26 | 23, 25 | eqtri 2765 | . . . . 5 ⊢ (𝐹‘(0(,)0)) = 〈0, 0〉 | 
| 27 | 26 | fveq2i 6909 | . . . 4 ⊢ ((,)‘(𝐹‘(0(,)0))) = ((,)‘〈0, 0〉) | 
| 28 | df-ov 7434 | . . . 4 ⊢ (0(,)0) = ((,)‘〈0, 0〉) | |
| 29 | 27, 28 | eqtr4i 2768 | . . 3 ⊢ ((,)‘(𝐹‘(0(,)0))) = (0(,)0) | 
| 30 | 24 | eqeq2i 2750 | . . . . . 6 ⊢ (𝐴 = (0(,)0) ↔ 𝐴 = ∅) | 
| 31 | 30 | biimpri 228 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = (0(,)0)) | 
| 32 | 31 | fveq2d 6910 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) = (𝐹‘(0(,)0))) | 
| 33 | 32 | fveq2d 6910 | . . 3 ⊢ (𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = ((,)‘(𝐹‘(0(,)0)))) | 
| 34 | 29, 33, 31 | 3eqtr4a 2803 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ∅c0 4333 ifcif 4525 𝒫 cpw 4600 〈cop 4632 ↦ cmpt 5225 × cxp 5683 ran crn 5686 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supcsup 9480 infcinf 9481 ℝcr 11154 0cc0 11155 ℝ*cxr 11294 < clt 11295 (,)cioo 13387 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-ioo 13391 | 
| This theorem is referenced by: uniioombllem2 25618 | 
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