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 6516 | . . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐷) |
3 | issmfdf.d | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | |
4 | 2, 3 | eqsstrd 4003 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
5 | 1 | ffdmd 6530 | . . 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 5816 | . . . . . . . . . 10 ⊢ Ⅎ𝑥dom 𝐹 |
10 | nfcv 2975 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝐷 | |
11 | 9, 10 | rabeqf 3480 | . . . . . . . . 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 7164 | . . . . . . . 8 ⊢ (𝜑 → (𝑆 ↾t dom 𝐹) = (𝑆 ↾t 𝐷)) |
14 | 12, 13 | eleq12d 2905 | . . . . . . 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 3214 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
19 | 4, 5, 18 | 3jca 1123 | . 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 2819 | . . 3 ⊢ dom 𝐹 = dom 𝐹 | |
22 | 8, 20, 21 | issmff 43002 | . 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 1082 = wceq 1531 Ⅎwnf 1778 ∈ wcel 2108 Ⅎwnfc 2959 ∀wral 3136 {crab 3140 ⊆ wss 3934 ∪ cuni 4830 class class class wbr 5057 dom cdm 5548 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ℝcr 10528 < clt 10667 ↾t crest 16686 SAlgcsalg 42584 SMblFncsmblfn 42968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-pre-lttri 10603 ax-pre-lttrn 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-er 8281 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-ioo 12734 df-ico 12736 df-smblfn 42969 |
This theorem is referenced by: issmfdmpt 43016 smfconst 43017 |
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