Theorem ismbf1 25601

Description: The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 25605 and ismbfcn 25606 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 5807	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (ℜ ∘ 𝑓) = (ℜ ∘ 𝐹))
2	1	cnveqd 5824	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡(ℜ ∘ 𝑓) = ◡(ℜ ∘ 𝐹))
3	2	imaeq1d 6018	. . . . 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 5807	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (ℑ ∘ 𝑓) = (ℑ ∘ 𝐹))
6	5	cnveqd 5824	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡(ℑ ∘ 𝑓) = ◡(ℑ ∘ 𝐹))
7	6	imaeq1d 6018	. . . . 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 25596	. 2 ⊢ MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
12	10, 11	elrab2 3638	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 5623  dom cdm 5624  ran crn 5625   “ cima 5627   ∘ ccom 5628  (class class class)co 7360   ↑_pm cpm 8767  ℂcc 11027  ℝcr 11028  (,)cioo 13289  ℜcre 15050  ℑcim 15051  volcvol 25440  MblFncmbf 25591

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-mbf 25596

This theorem is referenced by:  mbff  25602  mbfdm  25603  ismbf  25605  ismbfcn  25606  mbfconst  25610  mbfres  25621  cncombf  25635  cnmbf  25636  mbfdmssre  46446