mbff - Metamath Proof Explorer

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Theorem mbff 25673

Description: A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014.)

**Assertion**
Ref	Expression
mbff	⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)

Proof of Theorem mbff Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismbf1 25672	. . 3 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ∀𝑥 ∈ ran (,)((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
2	1	simplbi 497	. 2 ⊢ (𝐹 ∈ MblFn → 𝐹 ∈ (ℂ ↑_pm ℝ))
3		cnex 11233	. . . 4 ⊢ ℂ ∈ V
4		reex 11243	. . . 4 ⊢ ℝ ∈ V
5	3, 4	elpm2 8912	. . 3 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
6	5	simplbi 497	. 2 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → 𝐹:dom 𝐹⟶ℂ)
7	2, 6	syl 17	1 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ∀wral 3058 ⊆ wss 3962 ^◡ccnv 5687 dom cdm 5688 ran crn 5689 “ cima 5691 ∘ ccom 5692 ⟶wf 6558 (class class class)co 7430 ↑_pm cpm 8865 ℂcc 11150 ℝcr 11151 (,)cioo 13383 ℜcre 15132 ℑcim 15133 volcvol 25511 MblFncmbf 25662

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-pm 8867 df-mbf 25667

This theorem is referenced by: mbfdm 25674 mbfmptcl 25684 mbfres 25692 mbfimaopnlem 25703 mbfadd 25709 mbfsub 25710 mbfmul 25775 iblcnlem 25838 bddmulibl 25888 bddibl 25889 bddiblnc 25891 mbfresmf 46694

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