Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmfdf | Structured version Visualization version GIF version |
Description: A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
issmfdf.x | ⊢ Ⅎ𝑥𝐹 |
issmfdf.a | ⊢ Ⅎ𝑎𝜑 |
issmfdf.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
issmfdf.d | ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
issmfdf.f | ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
issmfdf.p | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) |
Ref | Expression |
---|---|
issmfdf | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmfdf.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) | |
2 | 1 | fdmd 6611 | . . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐷) |
3 | issmfdf.d | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | |
4 | 2, 3 | eqsstrd 3959 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
5 | 1 | ffdmd 6631 | . . 3 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
6 | issmfdf.a | . . . 4 ⊢ Ⅎ𝑎𝜑 | |
7 | issmfdf.p | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) | |
8 | issmfdf.x | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹 | |
9 | 8 | nfdm 5860 | . . . . . . . . . 10 ⊢ Ⅎ𝑥dom 𝐹 |
10 | nfcv 2907 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝐷 | |
11 | 9, 10 | rabeqf 3415 | . . . . . . . . 9 ⊢ (dom 𝐹 = 𝐷 → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎}) |
12 | 2, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎}) |
13 | 2 | oveq2d 7291 | . . . . . . . 8 ⊢ (𝜑 → (𝑆 ↾t dom 𝐹) = (𝑆 ↾t 𝐷)) |
14 | 12, 13 | eleq12d 2833 | . . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
15 | 14 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
16 | 7, 15 | mpbird 256 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
17 | 16 | ex 413 | . . . 4 ⊢ (𝜑 → (𝑎 ∈ ℝ → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹))) |
18 | 6, 17 | ralrimi 3141 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
19 | 4, 5, 18 | 3jca 1127 | . 2 ⊢ (𝜑 → (dom 𝐹 ⊆ ∪ 𝑆 ∧ 𝐹:dom 𝐹⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹))) |
20 | issmfdf.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
21 | eqid 2738 | . . 3 ⊢ dom 𝐹 = dom 𝐹 | |
22 | 8, 20, 21 | issmff 44270 | . 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (dom 𝐹 ⊆ ∪ 𝑆 ∧ 𝐹:dom 𝐹⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)))) |
23 | 19, 22 | mpbird 256 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2106 Ⅎwnfc 2887 ∀wral 3064 {crab 3068 ⊆ wss 3887 ∪ cuni 4839 class class class wbr 5074 dom cdm 5589 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 < clt 11009 ↾t crest 17131 SAlgcsalg 43849 SMblFncsmblfn 44233 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-ioo 13083 df-ico 13085 df-smblfn 44234 |
This theorem is referenced by: issmfdmpt 44284 smfconst 44285 |
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