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Theorem ismbf1 24154
Description: The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 24158 and ismbfcn 24159 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 5723 . . . . . . 7 (𝑓 = 𝐹 → (ℜ ∘ 𝑓) = (ℜ ∘ 𝐹))
21cnveqd 5740 . . . . . 6 (𝑓 = 𝐹(ℜ ∘ 𝑓) = (ℜ ∘ 𝐹))
32imaeq1d 5922 . . . . 5 (𝑓 = 𝐹 → ((ℜ ∘ 𝑓) “ 𝑥) = ((ℜ ∘ 𝐹) “ 𝑥))
43eleq1d 2897 . . . 4 (𝑓 = 𝐹 → (((ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ↔ ((ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
5 coeq2 5723 . . . . . . 7 (𝑓 = 𝐹 → (ℑ ∘ 𝑓) = (ℑ ∘ 𝐹))
65cnveqd 5740 . . . . . 6 (𝑓 = 𝐹(ℑ ∘ 𝑓) = (ℑ ∘ 𝐹))
76imaeq1d 5922 . . . . 5 (𝑓 = 𝐹 → ((ℑ ∘ 𝑓) “ 𝑥) = ((ℑ ∘ 𝐹) “ 𝑥))
87eleq1d 2897 . . . 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 3197 . 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 24149 . 2 MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)(((ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
1210, 11elrab2 3682 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 3138  ccnv 5548  dom cdm 5549  ran crn 5550  cima 5552  ccom 5553  (class class class)co 7145  pm cpm 8397  cc 10524  cr 10525  (,)cioo 12728  cre 14446  cim 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
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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-opab 5121  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-mbf 24149
This theorem is referenced by:  mbff  24155  mbfdm  24156  ismbf  24158  ismbfcn  24159  mbfconst  24163  mbfres  24174  cncombf  24188  cnmbf  24189  mbfdmssre  42166
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