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 6518 | . . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐷) |
3 | issmfdf.d | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | |
4 | 2, 3 | eqsstrd 4005 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
5 | 1 | ffdmd 6532 | . . 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 5818 | . . . . . . . . . 10 ⊢ Ⅎ𝑥dom 𝐹 |
10 | nfcv 2977 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝐷 | |
11 | 9, 10 | rabeqf 3482 | . . . . . . . . 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 7166 | . . . . . . . 8 ⊢ (𝜑 → (𝑆 ↾t dom 𝐹) = (𝑆 ↾t 𝐷)) |
14 | 12, 13 | eleq12d 2907 | . . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
15 | 14 | adantr 483 | . . . . . 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 259 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
17 | 16 | ex 415 | . . . 4 ⊢ (𝜑 → (𝑎 ∈ ℝ → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹))) |
18 | 6, 17 | ralrimi 3216 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
19 | 4, 5, 18 | 3jca 1124 | . 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 2821 | . . 3 ⊢ dom 𝐹 = dom 𝐹 | |
22 | 8, 20, 21 | issmff 43004 | . 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 259 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 Ⅎwnfc 2961 ∀wral 3138 {crab 3142 ⊆ wss 3936 ∪ cuni 4832 class class class wbr 5059 dom cdm 5550 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 < clt 10669 ↾t crest 16688 SAlgcsalg 42586 SMblFncsmblfn 42970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-ioo 12736 df-ico 12738 df-smblfn 42971 |
This theorem is referenced by: issmfdmpt 43018 smfconst 43019 |
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