Theorem ismbf1 25659

Description: The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 25663 and ismbfcn 25664 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 5869	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (ℜ ∘ 𝑓) = (ℜ ∘ 𝐹))
2	1	cnveqd 5886	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡(ℜ ∘ 𝑓) = ◡(ℜ ∘ 𝐹))
3	2	imaeq1d 6077	. . . . 5 ⊢ (𝑓 = 𝐹 → (◡(ℜ ∘ 𝑓) “ 𝑥) = (◡(ℜ ∘ 𝐹) “ 𝑥))
4	3	eleq1d 2826	. . . 4 ⊢ (𝑓 = 𝐹 → ((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ↔ (◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
5		coeq2 5869	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (ℑ ∘ 𝑓) = (ℑ ∘ 𝐹))
6	5	cnveqd 5886	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡(ℑ ∘ 𝑓) = ◡(ℑ ∘ 𝐹))
7	6	imaeq1d 6077	. . . . 5 ⊢ (𝑓 = 𝐹 → (◡(ℑ ∘ 𝑓) “ 𝑥) = (◡(ℑ ∘ 𝐹) “ 𝑥))
8	7	eleq1d 2826	. . . 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 3178	. 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 25654	. 2 ⊢ MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
12	10, 11	elrab2 3695	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 2108  ∀wral 3061  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   “ cima 5688   ∘ ccom 5689  (class class class)co 7431   ↑_pm cpm 8867  ℂcc 11153  ℝcr 11154  (,)cioo 13387  ℜcre 15136  ℑcim 15137  volcvol 25498  MblFncmbf 25649

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-mbf 25654

This theorem is referenced by:  mbff  25660  mbfdm  25661  ismbf  25663  ismbfcn  25664  mbfconst  25668  mbfres  25679  cncombf  25693  cnmbf  25694  mbfdmssre  46015