mbfdm - Metamath Proof Explorer

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Theorem mbfdm 24695

Description: The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)

**Assertion**
Ref	Expression
mbfdm	⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)

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

Step	Hyp	Ref	Expression
1		ref 14751	. . . 4 ⊢ ℜ:ℂ⟶ℝ
2		mbff 24694	. . . 4 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
3		fco 6608	. . . 4 ⊢ ((ℜ:ℂ⟶ℝ ∧ 𝐹:dom 𝐹⟶ℂ) → (ℜ ∘ 𝐹):dom 𝐹⟶ℝ)
4	1, 2, 3	sylancr 586	. . 3 ⊢ (𝐹 ∈ MblFn → (ℜ ∘ 𝐹):dom 𝐹⟶ℝ)
5		fimacnv 6606	. . 3 ⊢ ((ℜ ∘ 𝐹):dom 𝐹⟶ℝ → (^◡(ℜ ∘ 𝐹) “ ℝ) = dom 𝐹)
6	4, 5	syl 17	. 2 ⊢ (𝐹 ∈ MblFn → (^◡(ℜ ∘ 𝐹) “ ℝ) = dom 𝐹)
7		imaeq2 5954	. . . 4 ⊢ (𝑥 = ℝ → (^◡(ℜ ∘ 𝐹) “ 𝑥) = (^◡(ℜ ∘ 𝐹) “ ℝ))
8	7	eleq1d 2823	. . 3 ⊢ (𝑥 = ℝ → ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ↔ (^◡(ℜ ∘ 𝐹) “ ℝ) ∈ dom vol))
9		ismbf1 24693	. . . 4 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ∀𝑥 ∈ ran (,)((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
10		simpl 482	. . . . 5 ⊢ (((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol) → (^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)
11	10	ralimi 3086	. . . 4 ⊢ (∀𝑥 ∈ ran (,)((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol) → ∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)
12	9, 11	simplbiim 504	. . 3 ⊢ (𝐹 ∈ MblFn → ∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)
13		ioomax 13083	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
14		ioof 13108	. . . . . . 7 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
15		ffn 6584	. . . . . . 7 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
16	14, 15	ax-mp 5	. . . . . 6 ⊢ (,) Fn (ℝ^* × ℝ^*)
17		mnfxr 10963	. . . . . 6 ⊢ -∞ ∈ ℝ^*
18		pnfxr 10960	. . . . . 6 ⊢ +∞ ∈ ℝ^*
19		fnovrn 7425	. . . . . 6 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (-∞(,)+∞) ∈ ran (,))
20	16, 17, 18, 19	mp3an 1459	. . . . 5 ⊢ (-∞(,)+∞) ∈ ran (,)
21	13, 20	eqeltrri 2836	. . . 4 ⊢ ℝ ∈ ran (,)
22	21	a1i 11	. . 3 ⊢ (𝐹 ∈ MblFn → ℝ ∈ ran (,))
23	8, 12, 22	rspcdva 3554	. 2 ⊢ (𝐹 ∈ MblFn → (^◡(ℜ ∘ 𝐹) “ ℝ) ∈ dom vol)
24	6, 23	eqeltrrd 2840	1 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 𝒫 cpw 4530 × cxp 5578 ^◡ccnv 5579 dom cdm 5580 ran crn 5581 “ cima 5583 ∘ ccom 5584 Fn wfn 6413 ⟶wf 6414 (class class class)co 7255 ↑_pm cpm 8574 ℂcc 10800 ℝcr 10801 +∞cpnf 10937 -∞cmnf 10938 ℝ^*cxr 10939 (,)cioo 13008 ℜcre 14736 ℑcim 14737 volcvol 24532 MblFncmbf 24683

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-er 8456 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-2 11966 df-ioo 13012 df-cj 14738 df-re 14739 df-mbf 24688

This theorem is referenced by: ismbf 24697 ismbfcn 24698 mbfimaicc 24700 mbfdm2 24706 mbfres 24713 mbfmulc2lem 24716 mbfimaopn2 24726 cncombf 24727 mbfaddlem 24729 mbfadd 24730 mbfsub 24731 mbfmullem2 24794 mbfmul 24796 bddmulibl 24908 bddibl 24909 bddiblnc 24911 itgulm 25472 ftc1anclem1 35777 ftc1anclem5 35781 ftc1anclem8 35784 smfmbfcex 44182

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