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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 43012 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
5 | 1, 4 | mpbid 234 | . 2 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
6 | 5 | simp1d 1138 | 1 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 ⊆ wss 3938 ∪ cuni 4840 class class class wbr 5068 dom cdm 5557 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 < clt 10677 ↾t crest 16696 SAlgcsalg 42600 SMblFncsmblfn 42984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-ioo 12745 df-ico 12747 df-smblfn 42985 |
This theorem is referenced by: sssmf 43022 smfsssmf 43027 issmfle 43029 issmfgt 43040 smfadd 43048 issmfge 43053 smflim 43060 smfpimgtxr 43063 smfpimioo 43069 smfresal 43070 smfrec 43071 smfres 43072 smfmul 43077 smfmulc1 43078 smfco 43084 smfsuplem3 43094 |
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