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Mirrors > Home > MPE Home > Th. List > mbfdm | Structured version Visualization version GIF version |
Description: The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
Ref | Expression |
---|---|
mbfdm | ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ref 14461 | . . . 4 ⊢ ℜ:ℂ⟶ℝ | |
2 | mbff 24155 | . . . 4 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ) | |
3 | fco 6525 | . . . 4 ⊢ ((ℜ:ℂ⟶ℝ ∧ 𝐹:dom 𝐹⟶ℂ) → (ℜ ∘ 𝐹):dom 𝐹⟶ℝ) | |
4 | 1, 2, 3 | sylancr 587 | . . 3 ⊢ (𝐹 ∈ MblFn → (ℜ ∘ 𝐹):dom 𝐹⟶ℝ) |
5 | fimacnv 6832 | . . 3 ⊢ ((ℜ ∘ 𝐹):dom 𝐹⟶ℝ → (◡(ℜ ∘ 𝐹) “ ℝ) = dom 𝐹) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐹 ∈ MblFn → (◡(ℜ ∘ 𝐹) “ ℝ) = dom 𝐹) |
7 | imaeq2 5919 | . . . 4 ⊢ (𝑥 = ℝ → (◡(ℜ ∘ 𝐹) “ 𝑥) = (◡(ℜ ∘ 𝐹) “ ℝ)) | |
8 | 7 | eleq1d 2897 | . . 3 ⊢ (𝑥 = ℝ → ((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ↔ (◡(ℜ ∘ 𝐹) “ ℝ) ∈ dom vol)) |
9 | ismbf1 24154 | . . . 4 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))) | |
10 | simpl 483 | . . . . 5 ⊢ (((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol) → (◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol) | |
11 | 10 | ralimi 3160 | . . . 4 ⊢ (∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol) → ∀𝑥 ∈ ran (,)(◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol) |
12 | 9, 11 | simplbiim 505 | . . 3 ⊢ (𝐹 ∈ MblFn → ∀𝑥 ∈ ran (,)(◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol) |
13 | ioomax 12801 | . . . . 5 ⊢ (-∞(,)+∞) = ℝ | |
14 | ioof 12825 | . . . . . . 7 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
15 | ffn 6508 | . . . . . . 7 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 ⊢ (,) Fn (ℝ* × ℝ*) |
17 | mnfxr 10687 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
18 | pnfxr 10684 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
19 | fnovrn 7312 | . . . . . 6 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞(,)+∞) ∈ ran (,)) | |
20 | 16, 17, 18, 19 | mp3an 1452 | . . . . 5 ⊢ (-∞(,)+∞) ∈ ran (,) |
21 | 13, 20 | eqeltrri 2910 | . . . 4 ⊢ ℝ ∈ ran (,) |
22 | 21 | a1i 11 | . . 3 ⊢ (𝐹 ∈ MblFn → ℝ ∈ ran (,)) |
23 | 8, 12, 22 | rspcdva 3624 | . 2 ⊢ (𝐹 ∈ MblFn → (◡(ℜ ∘ 𝐹) “ ℝ) ∈ dom vol) |
24 | 6, 23 | eqeltrrd 2914 | 1 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3138 𝒫 cpw 4537 × cxp 5547 ◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 ∘ ccom 5553 Fn wfn 6344 ⟶wf 6345 (class class class)co 7145 ↑pm cpm 8397 ℂcc 10524 ℝcr 10525 +∞cpnf 10661 -∞cmnf 10662 ℝ*cxr 10663 (,)cioo 12728 ℜcre 14446 ℑcim 14447 volcvol 23993 MblFncmbf 24144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7680 df-2nd 7681 df-er 8279 df-pm 8399 df-en 8499 df-dom 8500 df-sdom 8501 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-2 11689 df-ioo 12732 df-cj 14448 df-re 14449 df-mbf 24149 |
This theorem is referenced by: ismbf 24158 ismbfcn 24159 mbfimaicc 24161 mbfdm2 24167 mbfres 24174 mbfmulc2lem 24177 mbfimaopn2 24187 cncombf 24188 mbfaddlem 24190 mbfadd 24191 mbfsub 24192 mbfmullem2 24254 mbfmul 24256 bddmulibl 24368 bddibl 24369 itgulm 24925 bddiblnc 34844 ftc1anclem1 34849 ftc1anclem5 34853 ftc1anclem8 34856 smfmbfcex 42917 |
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