Theorem ismbf1 25678

Description: The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 25682 and ismbfcn 25683 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 5883	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (ℜ ∘ 𝑓) = (ℜ ∘ 𝐹))
2	1	cnveqd 5900	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡(ℜ ∘ 𝑓) = ◡(ℜ ∘ 𝐹))
3	2	imaeq1d 6088	. . . . 5 ⊢ (𝑓 = 𝐹 → (◡(ℜ ∘ 𝑓) “ 𝑥) = (◡(ℜ ∘ 𝐹) “ 𝑥))
4	3	eleq1d 2829	. . . 4 ⊢ (𝑓 = 𝐹 → ((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ↔ (◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
5		coeq2 5883	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (ℑ ∘ 𝑓) = (ℑ ∘ 𝐹))
6	5	cnveqd 5900	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡(ℑ ∘ 𝑓) = ◡(ℑ ∘ 𝐹))
7	6	imaeq1d 6088	. . . . 5 ⊢ (𝑓 = 𝐹 → (◡(ℑ ∘ 𝑓) “ 𝑥) = (◡(ℑ ∘ 𝐹) “ 𝑥))
8	7	eleq1d 2829	. . . 4 ⊢ (𝑓 = 𝐹 → ((◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol ↔ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))
9	4, 8	anbi12d 631	. . 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 3184	. 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 25673	. 2 ⊢ MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
12	10, 11	elrab2 3711	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 1537   ∈ wcel 2108  ∀wral 3067  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   “ cima 5703   ∘ ccom 5704  (class class class)co 7448   ↑_pm cpm 8885  ℂcc 11182  ℝcr 11183  (,)cioo 13407  ℜcre 15146  ℑcim 15147  volcvol 25517  MblFncmbf 25668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-mbf 25673

This theorem is referenced by:  mbff  25679  mbfdm  25680  ismbf  25682  ismbfcn  25683  mbfconst  25687  mbfres  25698  cncombf  25712  cnmbf  25713  mbfdmssre  45921