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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 13364 | . . . 4 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
2 | ffn 6668 | . . . 4 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
3 | ovelrn 7530 | . . . 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 24939 | . . . . . . . 8 ⊢ ((𝑎(,)𝑏) ≠ ∅ → (𝐹‘(𝑎(,)𝑏)) = 〈𝑎, 𝑏〉) |
7 | 6 | fveq2d 6846 | . . . . . . 7 ⊢ ((𝑎(,)𝑏) ≠ ∅ → ((,)‘(𝐹‘(𝑎(,)𝑏))) = ((,)‘〈𝑎, 𝑏〉)) |
8 | df-ov 7360 | . . . . . . 7 ⊢ (𝑎(,)𝑏) = ((,)‘〈𝑎, 𝑏〉) | |
9 | 7, 8 | eqtr4di 2794 | . . . . . 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 284 | . . . . . . 7 ⊢ (𝐴 = (𝑎(,)𝑏) → (¬ 𝐴 = ∅ ↔ (𝑎(,)𝑏) ≠ ∅)) |
13 | 2fveq3 6847 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → ((,)‘(𝐹‘𝐴)) = ((,)‘(𝐹‘(𝑎(,)𝑏)))) | |
14 | id 22 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → 𝐴 = (𝑎(,)𝑏)) | |
15 | 13, 14 | eqeq12d 2752 | . . . . . . 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 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 3196 | . . 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 13368 | . . . . . . 7 ⊢ (0(,)0) ∈ ran (,) | |
22 | 5 | ioorval 24938 | . . . . . . 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 13292 | . . . . . . 7 ⊢ (0(,)0) = ∅ | |
25 | 24 | iftruei 4493 | . . . . . 6 ⊢ if((0(,)0) = ∅, 〈0, 0〉, 〈inf((0(,)0), ℝ*, < ), sup((0(,)0), ℝ*, < )〉) = 〈0, 0〉 |
26 | 23, 25 | eqtri 2764 | . . . . 5 ⊢ (𝐹‘(0(,)0)) = 〈0, 0〉 |
27 | 26 | fveq2i 6845 | . . . 4 ⊢ ((,)‘(𝐹‘(0(,)0))) = ((,)‘〈0, 0〉) |
28 | df-ov 7360 | . . . 4 ⊢ (0(,)0) = ((,)‘〈0, 0〉) | |
29 | 27, 28 | eqtr4i 2767 | . . 3 ⊢ ((,)‘(𝐹‘(0(,)0))) = (0(,)0) |
30 | 24 | eqeq2i 2749 | . . . . . 6 ⊢ (𝐴 = (0(,)0) ↔ 𝐴 = ∅) |
31 | 30 | biimpri 227 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = (0(,)0)) |
32 | 31 | fveq2d 6846 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) = (𝐹‘(0(,)0))) |
33 | 32 | fveq2d 6846 | . . 3 ⊢ (𝐴 = ∅ → ((,)‘(𝐹‘𝐴)) = ((,)‘(𝐹‘(0(,)0)))) |
34 | 29, 33, 31 | 3eqtr4a 2802 | . 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 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 ∅c0 4282 ifcif 4486 𝒫 cpw 4560 〈cop 4592 ↦ cmpt 5188 × cxp 5631 ran crn 5634 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 supcsup 9376 infcinf 9377 ℝcr 11050 0cc0 11051 ℝ*cxr 11188 < clt 11189 (,)cioo 13264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-q 12874 df-ioo 13268 |
This theorem is referenced by: uniioombllem2 24947 |
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