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Mirrors > Home > MPE Home > Th. List > mbfimaicc | Structured version Visualization version GIF version |
Description: The preimage of any closed interval under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
mbfimaicc | ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (◡𝐹 “ (𝐵[,]𝐶)) ∈ dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssre 12807 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵[,]𝐶) ⊆ ℝ) | |
2 | 1 | adantl 485 | . . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵[,]𝐶) ⊆ ℝ) |
3 | dfss4 4185 | . . . . . 6 ⊢ ((𝐵[,]𝐶) ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ (𝐵[,]𝐶))) = (𝐵[,]𝐶)) | |
4 | 2, 3 | sylib 221 | . . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (ℝ ∖ (ℝ ∖ (𝐵[,]𝐶))) = (𝐵[,]𝐶)) |
5 | difreicc 12862 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (ℝ ∖ (𝐵[,]𝐶)) = ((-∞(,)𝐵) ∪ (𝐶(,)+∞))) | |
6 | 5 | adantl 485 | . . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (ℝ ∖ (𝐵[,]𝐶)) = ((-∞(,)𝐵) ∪ (𝐶(,)+∞))) |
7 | 6 | difeq2d 4050 | . . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (ℝ ∖ (ℝ ∖ (𝐵[,]𝐶))) = (ℝ ∖ ((-∞(,)𝐵) ∪ (𝐶(,)+∞)))) |
8 | 4, 7 | eqtr3d 2835 | . . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵[,]𝐶) = (ℝ ∖ ((-∞(,)𝐵) ∪ (𝐶(,)+∞)))) |
9 | 8 | imaeq2d 5896 | . . 3 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (◡𝐹 “ (𝐵[,]𝐶)) = (◡𝐹 “ (ℝ ∖ ((-∞(,)𝐵) ∪ (𝐶(,)+∞))))) |
10 | ffun 6490 | . . . . . 6 ⊢ (𝐹:𝐴⟶ℝ → Fun 𝐹) | |
11 | funcnvcnv 6391 | . . . . . 6 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝐹:𝐴⟶ℝ → Fun ◡◡𝐹) |
13 | 12 | ad2antlr 726 | . . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → Fun ◡◡𝐹) |
14 | imadif 6408 | . . . 4 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (ℝ ∖ ((-∞(,)𝐵) ∪ (𝐶(,)+∞)))) = ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ ((-∞(,)𝐵) ∪ (𝐶(,)+∞))))) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (◡𝐹 “ (ℝ ∖ ((-∞(,)𝐵) ∪ (𝐶(,)+∞)))) = ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ ((-∞(,)𝐵) ∪ (𝐶(,)+∞))))) |
16 | 9, 15 | eqtrd 2833 | . 2 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (◡𝐹 “ (𝐵[,]𝐶)) = ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ ((-∞(,)𝐵) ∪ (𝐶(,)+∞))))) |
17 | fimacnv 6816 | . . . . . 6 ⊢ (𝐹:𝐴⟶ℝ → (◡𝐹 “ ℝ) = 𝐴) | |
18 | 17 | adantl 485 | . . . . 5 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (◡𝐹 “ ℝ) = 𝐴) |
19 | mbfdm 24230 | . . . . . 6 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol) | |
20 | fdm 6495 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → dom 𝐹 = 𝐴) | |
21 | 20 | eleq1d 2874 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℝ → (dom 𝐹 ∈ dom vol ↔ 𝐴 ∈ dom vol)) |
22 | 21 | biimpac 482 | . . . . . 6 ⊢ ((dom 𝐹 ∈ dom vol ∧ 𝐹:𝐴⟶ℝ) → 𝐴 ∈ dom vol) |
23 | 19, 22 | sylan 583 | . . . . 5 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → 𝐴 ∈ dom vol) |
24 | 18, 23 | eqeltrd 2890 | . . . 4 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (◡𝐹 “ ℝ) ∈ dom vol) |
25 | imaundi 5975 | . . . . 5 ⊢ (◡𝐹 “ ((-∞(,)𝐵) ∪ (𝐶(,)+∞))) = ((◡𝐹 “ (-∞(,)𝐵)) ∪ (◡𝐹 “ (𝐶(,)+∞))) | |
26 | mbfima 24234 | . . . . . 6 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (◡𝐹 “ (-∞(,)𝐵)) ∈ dom vol) | |
27 | mbfima 24234 | . . . . . 6 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (◡𝐹 “ (𝐶(,)+∞)) ∈ dom vol) | |
28 | unmbl 24141 | . . . . . 6 ⊢ (((◡𝐹 “ (-∞(,)𝐵)) ∈ dom vol ∧ (◡𝐹 “ (𝐶(,)+∞)) ∈ dom vol) → ((◡𝐹 “ (-∞(,)𝐵)) ∪ (◡𝐹 “ (𝐶(,)+∞))) ∈ dom vol) | |
29 | 26, 27, 28 | syl2anc 587 | . . . . 5 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → ((◡𝐹 “ (-∞(,)𝐵)) ∪ (◡𝐹 “ (𝐶(,)+∞))) ∈ dom vol) |
30 | 25, 29 | eqeltrid 2894 | . . . 4 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (◡𝐹 “ ((-∞(,)𝐵) ∪ (𝐶(,)+∞))) ∈ dom vol) |
31 | difmbl 24147 | . . . 4 ⊢ (((◡𝐹 “ ℝ) ∈ dom vol ∧ (◡𝐹 “ ((-∞(,)𝐵) ∪ (𝐶(,)+∞))) ∈ dom vol) → ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ ((-∞(,)𝐵) ∪ (𝐶(,)+∞)))) ∈ dom vol) | |
32 | 24, 30, 31 | syl2anc 587 | . . 3 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ ((-∞(,)𝐵) ∪ (𝐶(,)+∞)))) ∈ dom vol) |
33 | 32 | adantr 484 | . 2 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ ((-∞(,)𝐵) ∪ (𝐶(,)+∞)))) ∈ dom vol) |
34 | 16, 33 | eqeltrd 2890 | 1 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (◡𝐹 “ (𝐵[,]𝐶)) ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ∪ cun 3879 ⊆ wss 3881 ◡ccnv 5518 dom cdm 5519 “ cima 5522 Fun wfun 6318 ⟶wf 6320 (class class class)co 7135 ℝcr 10525 +∞cpnf 10661 -∞cmnf 10662 (,)cioo 12726 [,]cicc 12729 volcvol 24067 MblFncmbf 24218 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 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 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 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 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-xmet 20084 df-met 20085 df-ovol 24068 df-vol 24069 df-mbf 24223 |
This theorem is referenced by: mbfimasn 24236 |
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