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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmff | Structured version Visualization version GIF version |
Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
issmff.x | ⊢ Ⅎ𝑥𝐹 |
issmff.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
issmff.d | ⊢ 𝐷 = dom 𝐹 |
Ref | Expression |
---|---|
issmff | ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmff.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
2 | issmff.d | . . 3 ⊢ 𝐷 = dom 𝐹 | |
3 | 1, 2 | issmf 42466 | . 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
4 | nfcv 2927 | . . . . . . 7 ⊢ Ⅎ𝑦𝐷 | |
5 | issmff.x | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐹 | |
6 | 5 | nfdm 5664 | . . . . . . . 8 ⊢ Ⅎ𝑥dom 𝐹 |
7 | 2, 6 | nfcxfr 2925 | . . . . . . 7 ⊢ Ⅎ𝑥𝐷 |
8 | nfcv 2927 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑦 | |
9 | 5, 8 | nffv 6507 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
10 | nfcv 2927 | . . . . . . . 8 ⊢ Ⅎ𝑥 < | |
11 | nfcv 2927 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑎 | |
12 | 9, 10, 11 | nfbr 4973 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) < 𝑎 |
13 | nfv 1874 | . . . . . . 7 ⊢ Ⅎ𝑦(𝐹‘𝑥) < 𝑎 | |
14 | fveq2 6497 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) | |
15 | 14 | breq1d 4936 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) < 𝑎 ↔ (𝐹‘𝑥) < 𝑎)) |
16 | 4, 7, 12, 13, 15 | cbvrab 3406 | . . . . . 6 ⊢ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} |
17 | 16 | eleq1i 2851 | . . . . 5 ⊢ ({𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) |
18 | 17 | ralbii 3110 | . . . 4 ⊢ (∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) |
19 | 18 | 3anbi3i 1140 | . . 3 ⊢ ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
20 | 19 | a1i 11 | . 2 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
21 | 3, 20 | bitrd 271 | 1 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 Ⅎwnfc 2911 ∀wral 3083 {crab 3087 ⊆ wss 3824 ∪ cuni 4709 class class class wbr 4926 dom cdm 5404 ⟶wf 6182 ‘cfv 6186 (class class class)co 6975 ℝcr 10333 < clt 10473 ↾t crest 16549 SAlgcsalg 42054 SMblFncsmblfn 42438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-pre-lttri 10408 ax-pre-lttrn 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-po 5323 df-so 5324 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-ov 6978 df-oprab 6979 df-mpo 6980 df-1st 7500 df-2nd 7501 df-er 8088 df-pm 8208 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-ioo 12557 df-ico 12559 df-smblfn 42439 |
This theorem is referenced by: smfpreimaltf 42474 issmfdf 42475 smfpimltxr 42485 |
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