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Mirrors > Home > MPE Home > Th. List > ismbf1 | Structured version Visualization version Unicode version |
Description: The predicate " is a measurable function". This is more naturally stated for functions on the reals, see ismbf 23397 and ismbfcn 23398 for the decomposition of the real and imaginary parts. (Contributed by Mario Carneiro, 17-Jun-2014.) |
Ref | Expression |
---|---|
ismbf1 | MblFn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coeq2 5280 | . . . . . . 7 | |
2 | 1 | cnveqd 5298 | . . . . . 6 |
3 | 2 | imaeq1d 5465 | . . . . 5 |
4 | 3 | eleq1d 2686 | . . . 4 |
5 | coeq2 5280 | . . . . . . 7 | |
6 | 5 | cnveqd 5298 | . . . . . 6 |
7 | 6 | imaeq1d 5465 | . . . . 5 |
8 | 7 | eleq1d 2686 | . . . 4 |
9 | 4, 8 | anbi12d 747 | . . 3 |
10 | 9 | ralbidv 2986 | . 2 |
11 | df-mbf 23388 | . 2 MblFn | |
12 | 10, 11 | elrab2 3366 | 1 MblFn |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 ccnv 5113 cdm 5114 crn 5115 cima 5117 ccom 5118 (class class class)co 6650 cpm 7858 cc 9934 cr 9935 cioo 12175 cre 13837 cim 13838 cvol 23232 MblFncmbf 23383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-mbf 23388 |
This theorem is referenced by: mbff 23394 mbfdm 23395 ismbf 23397 ismbfcn 23398 mbfconst 23402 mbfres 23411 cncombf 23425 cnmbf 23426 mbfdmssre 40217 |
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