Theorem ismbf1 25525

Description: The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 25529 and ismbfcn 25530 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.

Step	Hyp	Ref	Expression
1		coeq2 5822	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (ℜ ∘ 𝑓) = (ℜ ∘ 𝐹))
2	1	cnveqd 5839	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡(ℜ ∘ 𝑓) = ◡(ℜ ∘ 𝐹))
3	2	imaeq1d 6030	. . . . 5 ⊢ (𝑓 = 𝐹 → (◡(ℜ ∘ 𝑓) “ 𝑥) = (◡(ℜ ∘ 𝐹) “ 𝑥))
4	3	eleq1d 2813	. . . 4 ⊢ (𝑓 = 𝐹 → ((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ↔ (◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
5		coeq2 5822	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (ℑ ∘ 𝑓) = (ℑ ∘ 𝐹))
6	5	cnveqd 5839	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡(ℑ ∘ 𝑓) = ◡(ℑ ∘ 𝐹))
7	6	imaeq1d 6030	. . . . 5 ⊢ (𝑓 = 𝐹 → (◡(ℑ ∘ 𝑓) “ 𝑥) = (◡(ℑ ∘ 𝐹) “ 𝑥))
8	7	eleq1d 2813	. . . 4 ⊢ (𝑓 = 𝐹 → ((◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol ↔ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))
9	4, 8	anbi12d 632	. . 3 ⊢ (𝑓 = 𝐹 → (((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol) ↔ ((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
10	9	ralbidv 3156	. 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 25520	. 2 ⊢ MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
12	10, 11	elrab2 3662	1 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   “ cima 5641   ∘ ccom 5642  (class class class)co 7387   ↑_pm cpm 8800  ℂcc 11066  ℝcr 11067  (,)cioo 13306  ℜcre 15063  ℑcim 15064  volcvol 25364  MblFncmbf 25515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-mbf 25520

This theorem is referenced by:  mbff  25526  mbfdm  25527  ismbf  25529  ismbfcn  25530  mbfconst  25534  mbfres  25545  cncombf  25559  cnmbf  25560  mbfdmssre  45998