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Mirrors > Home > MPE Home > Th. List > df-mbf | Structured version Visualization version Unicode version |
Description: Define the class of measurable functions on the reals. A real function is measurable if the preimage of every open interval is a measurable set (see ismbl 22529) and a complex function is measurable if the real and imaginary parts of the function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
Ref | Expression |
---|---|
df-mbf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmbf 22621 |
. 2
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2 | cre 13209 |
. . . . . . . . 9
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3 | vf |
. . . . . . . . . 10
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4 | 3 | cv 1454 |
. . . . . . . . 9
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5 | 2, 4 | ccom 4857 |
. . . . . . . 8
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6 | 5 | ccnv 4852 |
. . . . . . 7
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7 | vx |
. . . . . . . 8
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8 | 7 | cv 1454 |
. . . . . . 7
![]() ![]() |
9 | 6, 8 | cima 4856 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | cvol 22464 |
. . . . . . 7
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11 | 10 | cdm 4853 |
. . . . . 6
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12 | 9, 11 | wcel 1898 |
. . . . 5
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13 | cim 13210 |
. . . . . . . . 9
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14 | 13, 4 | ccom 4857 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() |
15 | 14 | ccnv 4852 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 15, 8 | cima 4856 |
. . . . . 6
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17 | 16, 11 | wcel 1898 |
. . . . 5
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18 | 12, 17 | wa 375 |
. . . 4
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19 | cioo 11664 |
. . . . 5
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20 | 19 | crn 4854 |
. . . 4
![]() ![]() ![]() |
21 | 18, 7, 20 | wral 2749 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | cc 9563 |
. . . 4
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23 | cr 9564 |
. . . 4
![]() ![]() | |
24 | cpm 7499 |
. . . 4
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25 | 22, 23, 24 | co 6315 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() |
26 | 21, 3, 25 | crab 2753 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 1, 26 | wceq 1455 |
1
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Colors of variables: wff setvar class |
This definition is referenced by: ismbf1 22631 |
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