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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smff | Structured version Visualization version GIF version |
Description: A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smff.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smff.f | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
smff.d | ⊢ 𝐷 = dom 𝐹 |
Ref | Expression |
---|---|
smff | ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smff.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | |
2 | smff.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | smff.d | . . . 4 ⊢ 𝐷 = dom 𝐹 | |
4 | 2, 3 | issmf 46254 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
5 | 1, 4 | mpbid 231 | . 2 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
6 | 5 | simp2d 1140 | 1 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 {crab 3418 ⊆ wss 3944 ∪ cuni 4909 class class class wbr 5149 dom cdm 5678 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℝcr 11139 < clt 11280 ↾t crest 17405 SAlgcsalg 45834 SMblFncsmblfn 46221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-pre-lttri 11214 ax-pre-lttrn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-ioo 13363 df-ico 13365 df-smblfn 46222 |
This theorem is referenced by: sssmf 46264 smfsssmf 46269 issmfle 46271 smfpimltxr 46273 issmfgt 46282 issmfge 46296 smflimlem2 46298 smflimlem3 46299 smflimlem4 46300 smflim 46303 smfpimgtxr 46306 smfpimioompt 46312 smfpimioo 46313 smfresal 46314 smfres 46316 smfco 46328 smffmptf 46330 smfsuplem1 46337 smfsuplem3 46339 smfsupxr 46342 smfinflem 46343 smflimsuplem2 46347 smflimsuplem3 46348 smflimsuplem4 46349 smflimsuplem5 46350 smfliminflem 46356 smfpimne 46365 smfpimne2 46366 smfsupdmmbllem 46370 smfinfdmmbllem 46374 |
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