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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 41678 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
5 | 1, 4 | mpbid 224 | . 2 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
6 | 5 | simp2d 1174 | 1 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ∀wral 3090 {crab 3094 ⊆ wss 3770 ∪ cuni 4629 class class class wbr 4844 dom cdm 5313 ⟶wf 6098 ‘cfv 6102 (class class class)co 6879 ℝcr 10224 < clt 10364 ↾t crest 16395 SAlgcsalg 41266 SMblFncsmblfn 41650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-pre-lttri 10299 ax-pre-lttrn 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-po 5234 df-so 5235 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-1st 7402 df-2nd 7403 df-er 7983 df-pm 8099 df-en 8197 df-dom 8198 df-sdom 8199 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-ioo 12427 df-ico 12429 df-smblfn 41651 |
This theorem is referenced by: sssmf 41688 smfsssmf 41693 issmfle 41695 issmfgt 41706 issmfge 41719 smflimlem2 41721 smflimlem3 41722 smflimlem4 41723 smflim 41726 smfpimgtxr 41729 smfpimioompt 41734 smfpimioo 41735 smfresal 41736 smfres 41738 smfco 41750 smffmpt 41752 smfsuplem1 41758 smfsuplem3 41760 smfsupxr 41763 smfinflem 41764 smflimsuplem2 41768 smflimsuplem3 41769 smflimsuplem4 41770 smflimsuplem5 41771 smfliminflem 41777 |
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