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Theorem ismbf1 24152
Description: The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 24156 and ismbfcn 24157 for the decomposition of the real and imaginary parts. (Contributed by Mario Carneiro, 17-Jun-2014.)
Assertion
Ref Expression
ismbf1 (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)(((ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
Distinct variable group:   𝑥,𝐹

Proof of Theorem ismbf1
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 coeq2 5722 . . . . . . 7 (𝑓 = 𝐹 → (ℜ ∘ 𝑓) = (ℜ ∘ 𝐹))
21cnveqd 5739 . . . . . 6 (𝑓 = 𝐹(ℜ ∘ 𝑓) = (ℜ ∘ 𝐹))
32imaeq1d 5921 . . . . 5 (𝑓 = 𝐹 → ((ℜ ∘ 𝑓) “ 𝑥) = ((ℜ ∘ 𝐹) “ 𝑥))
43eleq1d 2894 . . . 4 (𝑓 = 𝐹 → (((ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ↔ ((ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
5 coeq2 5722 . . . . . . 7 (𝑓 = 𝐹 → (ℑ ∘ 𝑓) = (ℑ ∘ 𝐹))
65cnveqd 5739 . . . . . 6 (𝑓 = 𝐹(ℑ ∘ 𝑓) = (ℑ ∘ 𝐹))
76imaeq1d 5921 . . . . 5 (𝑓 = 𝐹 → ((ℑ ∘ 𝑓) “ 𝑥) = ((ℑ ∘ 𝐹) “ 𝑥))
87eleq1d 2894 . . . 4 (𝑓 = 𝐹 → (((ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol ↔ ((ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))
94, 8anbi12d 630 . . 3 (𝑓 = 𝐹 → ((((ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol) ↔ (((ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
109ralbidv 3194 . 2 (𝑓 = 𝐹 → (∀𝑥 ∈ ran (,)(((ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol) ↔ ∀𝑥 ∈ ran (,)(((ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
11 df-mbf 24147 . 2 MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)(((ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
1210, 11elrab2 3680 1 (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)(((ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  ccom 5552  (class class class)co 7145  pm cpm 8396  cc 10523  cr 10524  (,)cioo 12726  cre 14444  cim 14445  volcvol 23991  MblFncmbf 24142
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 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-mbf 24147
This theorem is referenced by:  mbff  24153  mbfdm  24154  ismbf  24156  ismbfcn  24157  mbfconst  24161  mbfres  24172  cncombf  24186  cnmbf  24187  mbfdmssre  42162
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