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Theorem ismbf1 25673
Description: The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 25677 and ismbfcn 25678 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 5872 . . . . . . 7 (𝑓 = 𝐹 → (ℜ ∘ 𝑓) = (ℜ ∘ 𝐹))
21cnveqd 5889 . . . . . 6 (𝑓 = 𝐹(ℜ ∘ 𝑓) = (ℜ ∘ 𝐹))
32imaeq1d 6079 . . . . 5 (𝑓 = 𝐹 → ((ℜ ∘ 𝑓) “ 𝑥) = ((ℜ ∘ 𝐹) “ 𝑥))
43eleq1d 2824 . . . 4 (𝑓 = 𝐹 → (((ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ↔ ((ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
5 coeq2 5872 . . . . . . 7 (𝑓 = 𝐹 → (ℑ ∘ 𝑓) = (ℑ ∘ 𝐹))
65cnveqd 5889 . . . . . 6 (𝑓 = 𝐹(ℑ ∘ 𝑓) = (ℑ ∘ 𝐹))
76imaeq1d 6079 . . . . 5 (𝑓 = 𝐹 → ((ℑ ∘ 𝑓) “ 𝑥) = ((ℑ ∘ 𝐹) “ 𝑥))
87eleq1d 2824 . . . 4 (𝑓 = 𝐹 → (((ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol ↔ ((ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))
94, 8anbi12d 632 . . 3 (𝑓 = 𝐹 → ((((ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol) ↔ (((ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
109ralbidv 3176 . 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 25668 . 2 MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)(((ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
1210, 11elrab2 3698 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 206  wa 395   = wceq 1537  wcel 2106  wral 3059  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  ccom 5693  (class class class)co 7431  pm cpm 8866  cc 11151  cr 11152  (,)cioo 13384  cre 15133  cim 15134  volcvol 25512  MblFncmbf 25663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-mbf 25668
This theorem is referenced by:  mbff  25674  mbfdm  25675  ismbf  25677  ismbfcn  25678  mbfconst  25682  mbfres  25693  cncombf  25707  cnmbf  25708  mbfdmssre  45956
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