mbff - Metamath Proof Explorer

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Theorem mbff 25752

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 25751	. . 3 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ∀𝑥 ∈ ran (,)((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
2	1	simplbi 501	. 2 ⊢ (𝐹 ∈ MblFn → 𝐹 ∈ (ℂ ↑_pm ℝ))
3		cnex 11180	. . . 4 ⊢ ℂ ∈ V
4		reex 11190	. . . 4 ⊢ ℝ ∈ V
5	3, 4	elpm2 8871	. . 3 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
6	5	simplbi 501	. 2 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → 𝐹:dom 𝐹⟶ℂ)
7	2, 6	syl 18	1 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∈ wcel 2149 ∀wral 3085 ⊆ wss 3913 ^◡ccnv 5661 dom cdm 5662 ran crn 5663 “ cima 5665 ∘ ccom 5666 ⟶wf 6533 (class class class)co 7411 ↑_pm cpm 8824 ℂcc 11097 ℝcr 11098 (,)cioo 13371 ℜcre 15147 ℑcim 15148 volcvol 25590 MblFncmbf 25741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-sbc 3754 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-opab 5178 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7414 df-oprab 7415 df-mpo 7416 df-pm 8826 df-mbf 25746

This theorem is referenced by: mbfdm 25753 mbfmptcl 25763 mbfres 25771 mbfimaopnlem 25782 mbfadd 25788 mbfsub 25789 mbfmul 25853 iblcnlem 25916 bddmulibl 25966 bddibl 25967 bddiblnc 25969 mbfresmf 47344

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