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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfdmss | Structured version Visualization version GIF version |
Description: The domain of a function measurable w.r.t. to a sigma-algebra, is a subset of the set underlying the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfdmss.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfdmss.f | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
smfdmss.d | ⊢ 𝐷 = dom 𝐹 |
Ref | Expression |
---|---|
smfdmss | ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfdmss.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | |
2 | smfdmss.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | smfdmss.d | . . . 4 ⊢ 𝐷 = dom 𝐹 | |
4 | 2, 3 | issmf 41864 | . . 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 | simp1d 1133 | 1 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∀wral 3090 {crab 3094 ⊆ wss 3792 ∪ cuni 4671 class class class wbr 4886 dom cdm 5355 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 < clt 10411 ↾t crest 16467 SAlgcsalg 41452 SMblFncsmblfn 41836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-pre-lttri 10346 ax-pre-lttrn 10347 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-er 8026 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-ioo 12491 df-ico 12493 df-smblfn 41837 |
This theorem is referenced by: sssmf 41874 smfsssmf 41879 issmfle 41881 issmfgt 41892 smfadd 41900 issmfge 41905 smflim 41912 smfpimgtxr 41915 smfpimioo 41921 smfresal 41922 smfrec 41923 smfres 41924 smfmul 41929 smfmulc1 41930 smfco 41936 smfsuplem3 41946 |
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