mbff - Metamath Proof Explorer

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Theorem mbff 25610

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 25609	. . 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 11110	. . . 4 ⊢ ℂ ∈ V
4		reex 11120	. . . 4 ⊢ ℝ ∈ V
5	3, 4	elpm2 8812	. . 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 396 ∈ wcel 2119 ∀wral 3053 ⊆ wss 3883 ^◡ccnv 5617 dom cdm 5618 ran crn 5619 “ cima 5621 ∘ ccom 5622 ⟶wf 6481 (class class class)co 7356 ↑_pm cpm 8764 ℂcc 11027 ℝcr 11028 (,)cioo 13289 ℜcre 15050 ℑcim 15051 volcvol 25448 MblFncmbf 25599

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-sbc 3724 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-br 5073 df-opab 5135 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-fv 6493 df-ov 7359 df-oprab 7360 df-mpo 7361 df-pm 8766 df-mbf 25604

This theorem is referenced by: mbfdm 25611 mbfmptcl 25621 mbfres 25629 mbfimaopnlem 25640 mbfadd 25646 mbfsub 25647 mbfmul 25711 iblcnlem 25774 bddmulibl 25824 bddibl 25825 bddiblnc 25827 mbfresmf 47182

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