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| Mirrors > Home > MPE Home > Th. List > Mathboxes > issmfled | Structured version Visualization version GIF version | ||
| Description: A sufficient condition for "𝐹 being a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
| Ref | Expression |
|---|---|
| issmfled.a | ⊢ Ⅎ𝑎𝜑 |
| issmfled.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
| issmfled.d | ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
| issmfled.f | ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
| issmfled.6 | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)) |
| Ref | Expression |
|---|---|
| issmfled | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | issmfled.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) | |
| 2 | 1 | fdmd 6709 | . . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐷) |
| 3 | issmfled.d | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | |
| 4 | 2, 3 | eqsstrd 3965 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
| 5 | 1 | ffdmd 6729 | . . 3 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
| 6 | issmfled.a | . . . 4 ⊢ Ⅎ𝑎𝜑 | |
| 7 | issmfled.6 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)) | |
| 8 | 2 | rabeqdv 3427 | . . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≤ 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎}) |
| 9 | 2 | oveq2d 7425 | . . . . . . . 8 ⊢ (𝜑 → (𝑆 ↾t dom 𝐹) = (𝑆 ↾t 𝐷)) |
| 10 | 8, 9 | eleq12d 2854 | . . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t dom 𝐹) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷))) |
| 11 | 10 | adantr 486 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t dom 𝐹) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷))) |
| 12 | 7, 11 | mpbird 260 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
| 13 | 12 | ex 418 | . . . 4 ⊢ (𝜑 → (𝑎 ∈ ℝ → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t dom 𝐹))) |
| 14 | 6, 13 | ralrimi 3260 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
| 15 | 4, 5, 14 | 3jca 1146 | . 2 ⊢ (𝜑 → (dom 𝐹 ⊆ ∪ 𝑆 ∧ 𝐹:dom 𝐹⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t dom 𝐹))) |
| 16 | issmfled.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
| 17 | eqid 2760 | . . 3 ⊢ dom 𝐹 = dom 𝐹 | |
| 18 | 16, 17 | issmfle 47671 | . 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (dom 𝐹 ⊆ ∪ 𝑆 ∧ 𝐹:dom 𝐹⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t dom 𝐹)))) |
| 19 | 15, 18 | mpbird 260 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 Ⅎwnf 1816 ∈ wcel 2145 ∀wral 3076 {crab 3412 ⊆ wss 3899 ∪ cuni 4867 class class class wbr 5103 dom cdm 5648 ⟶wf 6524 ‘cfv 6528 (class class class)co 7409 ℝcr 11156 ≤ cle 11301 ↾t crest 17538 SAlgcsalg 47234 SMblFncsmblfn 47621 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-inf2 9620 ax-cc 10470 ax-cnex 11213 ax-resscn 11214 ax-1cn 11215 ax-icn 11216 ax-addcl 11217 ax-addrcl 11218 ax-mulcl 11219 ax-mulrcl 11220 ax-mulcom 11221 ax-addass 11222 ax-mulass 11223 ax-distr 11224 ax-i2m1 11225 ax-1ne0 11226 ax-1rid 11227 ax-rnegex 11228 ax-rrecex 11229 ax-cnre 11230 ax-pre-lttri 11231 ax-pre-lttrn 11232 ax-pre-ltadd 11233 ax-pre-mulgt0 11234 ax-pre-sup 11235 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6294 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-isom 6537 df-riota 7366 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8455 df-er 8696 df-map 8828 df-pm 8829 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-card 9977 df-acn 9980 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11500 df-neg 11501 df-div 11929 df-nn 12291 df-n0 12562 df-z 12649 df-uz 12921 df-q 13031 df-rp 13076 df-ioo 13435 df-ico 13437 df-fl 13886 df-rest 17540 df-salg 47235 df-smblfn 47622 |
| This theorem is used by: smflim 47703 issmfle2d 47735 |
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