mbfdm - Metamath Proof Explorer

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Theorem mbfdm 23914

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 14309	. . . 4 ⊢ ℜ:ℂ⟶ℝ
2		mbff 23913	. . . 4 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
3		fco 6406	. . . 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 6711	. . 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 5809	. . . 4 ⊢ (𝑥 = ℝ → (^◡(ℜ ∘ 𝐹) “ 𝑥) = (^◡(ℜ ∘ 𝐹) “ ℝ))
8	7	eleq1d 2869	. . 3 ⊢ (𝑥 = ℝ → ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ↔ (^◡(ℜ ∘ 𝐹) “ ℝ) ∈ dom vol))
9		ismbf1 23912	. . . 4 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ∀𝑥 ∈ ran (,)((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
10		simpl 483	. . . . 5 ⊢ (((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol) → (^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)
11	10	ralimi 3129	. . . 4 ⊢ (∀𝑥 ∈ ran (,)((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol) → ∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)
12	9, 11	simplbiim 505	. . 3 ⊢ (𝐹 ∈ MblFn → ∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)
13		ioomax 12665	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
14		ioof 12689	. . . . . . 7 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
15		ffn 6389	. . . . . . 7 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
16	14, 15	ax-mp 5	. . . . . 6 ⊢ (,) Fn (ℝ^* × ℝ^*)
17		mnfxr 10551	. . . . . 6 ⊢ -∞ ∈ ℝ^*
18		pnfxr 10548	. . . . . 6 ⊢ +∞ ∈ ℝ^*
19		fnovrn 7186	. . . . . 6 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (-∞(,)+∞) ∈ ran (,))
20	16, 17, 18, 19	mp3an 1453	. . . . 5 ⊢ (-∞(,)+∞) ∈ ran (,)
21	13, 20	eqeltrri 2882	. . . 4 ⊢ ℝ ∈ ran (,)
22	21	a1i 11	. . 3 ⊢ (𝐹 ∈ MblFn → ℝ ∈ ran (,))
23	8, 12, 22	rspcdva 3567	. 2 ⊢ (𝐹 ∈ MblFn → (^◡(ℜ ∘ 𝐹) “ ℝ) ∈ dom vol)
24	6, 23	eqeltrrd 2886	1 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∀wral 3107 𝒫 cpw 4459 × cxp 5448 ^◡ccnv 5449 dom cdm 5450 ran crn 5451 “ cima 5453 ∘ ccom 5454 Fn wfn 6227 ⟶wf 6228 (class class class)co 7023 ↑_pm cpm 8264 ℂcc 10388 ℝcr 10389 +∞cpnf 10525 -∞cmnf 10526 ℝ^*cxr 10527 (,)cioo 12592 ℜcre 14294 ℑcim 14295 volcvol 23751 MblFncmbf 23902

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-po 5369 df-so 5370 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-1st 7552 df-2nd 7553 df-er 8146 df-pm 8266 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-2 11554 df-ioo 12596 df-cj 14296 df-re 14297 df-mbf 23907

This theorem is referenced by: ismbf 23916 ismbfcn 23917 mbfimaicc 23919 mbfdm2 23925 mbfres 23932 mbfmulc2lem 23935 mbfimaopn2 23945 cncombf 23946 mbfaddlem 23948 mbfadd 23949 mbfsub 23950 mbfmullem2 24012 mbfmul 24014 bddmulibl 24126 bddibl 24127 itgulm 24683 bddiblnc 34514 ftc1anclem1 34519 ftc1anclem5 34523 ftc1anclem8 34526 smfmbfcex 42600

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