Theorem ismbf1 25593

Description: The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 25597 and ismbfcn 25598 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 5815	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (ℜ ∘ 𝑓) = (ℜ ∘ 𝐹))
2	1	cnveqd 5832	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡(ℜ ∘ 𝑓) = ◡(ℜ ∘ 𝐹))
3	2	imaeq1d 6026	. . . . 5 ⊢ (𝑓 = 𝐹 → (◡(ℜ ∘ 𝑓) “ 𝑥) = (◡(ℜ ∘ 𝐹) “ 𝑥))
4	3	eleq1d 2822	. . . 4 ⊢ (𝑓 = 𝐹 → ((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ↔ (◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
5		coeq2 5815	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (ℑ ∘ 𝑓) = (ℑ ∘ 𝐹))
6	5	cnveqd 5832	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡(ℑ ∘ 𝑓) = ◡(ℑ ∘ 𝐹))
7	6	imaeq1d 6026	. . . . 5 ⊢ (𝑓 = 𝐹 → (◡(ℑ ∘ 𝑓) “ 𝑥) = (◡(ℑ ∘ 𝐹) “ 𝑥))
8	7	eleq1d 2822	. . . 4 ⊢ (𝑓 = 𝐹 → ((◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol ↔ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))
9	4, 8	anbi12d 633	. . 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 3161	. 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 25588	. 2 ⊢ MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
12	10, 11	elrab2 3651	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 1542   ∈ wcel 2114  ∀wral 3052  ^◡ccnv 5631  dom cdm 5632  ran crn 5633   “ cima 5635   ∘ ccom 5636  (class class class)co 7368   ↑_pm cpm 8776  ℂcc 11036  ℝcr 11037  (,)cioo 13273  ℜcre 15032  ℑcim 15033  volcvol 25432  MblFncmbf 25583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-mbf 25588

This theorem is referenced by:  mbff  25594  mbfdm  25595  ismbf  25597  ismbfcn  25598  mbfconst  25602  mbfres  25613  cncombf  25627  cnmbf  25628  mbfdmssre  46358