mbfdm - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mbfdm		Structured version Visualization version GIF version

Theorem mbfdm 25661

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 15151	. . . 4 ⊢ ℜ:ℂ⟶ℝ
2		mbff 25660	. . . 4 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
3		fco 6760	. . . 4 ⊢ ((ℜ:ℂ⟶ℝ ∧ 𝐹:dom 𝐹⟶ℂ) → (ℜ ∘ 𝐹):dom 𝐹⟶ℝ)
4	1, 2, 3	sylancr 587	. . 3 ⊢ (𝐹 ∈ MblFn → (ℜ ∘ 𝐹):dom 𝐹⟶ℝ)
5		fimacnv 6758	. . 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 6074	. . . 4 ⊢ (𝑥 = ℝ → (^◡(ℜ ∘ 𝐹) “ 𝑥) = (^◡(ℜ ∘ 𝐹) “ ℝ))
8	7	eleq1d 2826	. . 3 ⊢ (𝑥 = ℝ → ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ↔ (^◡(ℜ ∘ 𝐹) “ ℝ) ∈ dom vol))
9		ismbf1 25659	. . . 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 3083	. . . 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 13462	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
14		ioof 13487	. . . . . . 7 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
15		ffn 6736	. . . . . . 7 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
16	14, 15	ax-mp 5	. . . . . 6 ⊢ (,) Fn (ℝ^* × ℝ^*)
17		mnfxr 11318	. . . . . 6 ⊢ -∞ ∈ ℝ^*
18		pnfxr 11315	. . . . . 6 ⊢ +∞ ∈ ℝ^*
19		fnovrn 7608	. . . . . 6 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (-∞(,)+∞) ∈ ran (,))
20	16, 17, 18, 19	mp3an 1463	. . . . 5 ⊢ (-∞(,)+∞) ∈ ran (,)
21	13, 20	eqeltrri 2838	. . . 4 ⊢ ℝ ∈ ran (,)
22	21	a1i 11	. . 3 ⊢ (𝐹 ∈ MblFn → ℝ ∈ ran (,))
23	8, 12, 22	rspcdva 3623	. 2 ⊢ (𝐹 ∈ MblFn → (^◡(ℜ ∘ 𝐹) “ ℝ) ∈ dom vol)
24	6, 23	eqeltrrd 2842	1 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 𝒫 cpw 4600 × cxp 5683 ^◡ccnv 5684 dom cdm 5685 ran crn 5686 “ cima 5688 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 (class class class)co 7431 ↑_pm cpm 8867 ℂcc 11153 ℝcr 11154 +∞cpnf 11292 -∞cmnf 11293 ℝ^*cxr 11294 (,)cioo 13387 ℜcre 15136 ℑcim 15137 volcvol 25498 MblFncmbf 25649

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-ioo 13391 df-cj 15138 df-re 15139 df-mbf 25654

This theorem is referenced by: ismbf 25663 ismbfcn 25664 mbfimaicc 25666 mbfdm2 25672 mbfres 25679 mbfmulc2lem 25682 mbfimaopn2 25692 cncombf 25693 mbfaddlem 25695 mbfadd 25696 mbfsub 25697 mbfmullem2 25759 mbfmul 25761 bddmulibl 25874 bddibl 25875 bddiblnc 25877 itgulm 26451 ftc1anclem1 37700 ftc1anclem5 37704 ftc1anclem8 37707 smfmbfcex 46775

W3C validator