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Theorem mbfdm 25674

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 15147	. . . 4 ⊢ ℜ:ℂ⟶ℝ
2		mbff 25673	. . . 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 6075	. . . 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 25672	. . . 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 3080	. . . 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 13458	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
14		ioof 13483	. . . . . . 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 11315	. . . . . 6 ⊢ -∞ ∈ ℝ^*
18		pnfxr 11312	. . . . . 6 ⊢ +∞ ∈ ℝ^*
19		fnovrn 7607	. . . . . 6 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (-∞(,)+∞) ∈ ran (,))
20	16, 17, 18, 19	mp3an 1460	. . . . 5 ⊢ (-∞(,)+∞) ∈ ran (,)
21	13, 20	eqeltrri 2835	. . . 4 ⊢ ℝ ∈ ran (,)
22	21	a1i 11	. . 3 ⊢ (𝐹 ∈ MblFn → ℝ ∈ ran (,))
23	8, 12, 22	rspcdva 3622	. 2 ⊢ (𝐹 ∈ MblFn → (^◡(ℜ ∘ 𝐹) “ ℝ) ∈ dom vol)
24	6, 23	eqeltrrd 2839	1 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 𝒫 cpw 4604 × cxp 5686 ^◡ccnv 5687 dom cdm 5688 ran crn 5689 “ cima 5691 ∘ ccom 5692 Fn wfn 6557 ⟶wf 6558 (class class class)co 7430 ↑_pm cpm 8865 ℂcc 11150 ℝcr 11151 +∞cpnf 11289 -∞cmnf 11290 ℝ^*cxr 11291 (,)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 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 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-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 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-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-er 8743 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-2 12326 df-ioo 13387 df-cj 15134 df-re 15135 df-mbf 25667

This theorem is referenced by: ismbf 25676 ismbfcn 25677 mbfimaicc 25679 mbfdm2 25685 mbfres 25692 mbfmulc2lem 25695 mbfimaopn2 25705 cncombf 25706 mbfaddlem 25708 mbfadd 25709 mbfsub 25710 mbfmullem2 25773 mbfmul 25775 bddmulibl 25888 bddibl 25889 bddiblnc 25891 itgulm 26465 ftc1anclem1 37679 ftc1anclem5 37683 ftc1anclem8 37686 smfmbfcex 46715

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