| Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > issmfd | 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 |
|---|---|
| issmfd.a | ⊢ Ⅎ𝑎𝜑 |
| issmfd.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
| issmfd.d | ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
| issmfd.f | ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
| issmfd.p | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) |
| Ref | Expression |
|---|---|
| issmfd | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | issmfd.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) | |
| 2 | 1 | fdmd 6746 | . . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐷) |
| 3 | issmfd.d | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | |
| 4 | 2, 3 | eqsstrd 4018 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
| 5 | 1 | ffdmd 6766 | . . 3 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
| 6 | issmfd.a | . . . 4 ⊢ Ⅎ𝑎𝜑 | |
| 7 | issmfd.p | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) | |
| 8 | 2 | rabeqdv 3452 | . . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎}) |
| 9 | 2 | oveq2d 7447 | . . . . . . . 8 ⊢ (𝜑 → (𝑆 ↾t dom 𝐹) = (𝑆 ↾t 𝐷)) |
| 10 | 8, 9 | eleq12d 2835 | . . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
| 11 | 10 | adantr 480 | . . . . . 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 257 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
| 13 | 12 | ex 412 | . . . 4 ⊢ (𝜑 → (𝑎 ∈ ℝ → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹))) |
| 14 | 6, 13 | ralrimi 3257 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
| 15 | 4, 5, 14 | 3jca 1129 | . 2 ⊢ (𝜑 → (dom 𝐹 ⊆ ∪ 𝑆 ∧ 𝐹:dom 𝐹⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹))) |
| 16 | issmfd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
| 17 | eqid 2737 | . . 3 ⊢ dom 𝐹 = dom 𝐹 | |
| 18 | 16, 17 | issmf 46743 | . 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 257 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 < clt 11295 ↾t crest 17465 SAlgcsalg 46323 SMblFncsmblfn 46710 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 df-ico 13393 df-smblfn 46711 |
| This theorem is referenced by: sssmf 46753 mbfresmf 46754 cnfsmf 46755 incsmf 46757 smfsssmf 46758 smfres 46805 smfco 46817 |
| Copyright terms: Public domain | W3C validator |