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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 46748 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) | 
| 5 | 1, 4 | mpbid 232 | . 2 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) | 
| 6 | 5 | simp2d 1143 | 1 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 {crab 3435 ⊆ wss 3950 ∪ cuni 4906 class class class wbr 5142 dom cdm 5684 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 < clt 11296 ↾t crest 17466 SAlgcsalg 46328 SMblFncsmblfn 46715 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 ax-pre-lttrn 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-er 8746 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-ioo 13392 df-ico 13394 df-smblfn 46716 | 
| This theorem is referenced by: sssmf 46758 smfsssmf 46763 issmfle 46765 smfpimltxr 46767 issmfgt 46776 issmfge 46790 smflimlem2 46792 smflimlem3 46793 smflimlem4 46794 smflim 46797 smfpimgtxr 46800 smfpimioompt 46806 smfpimioo 46807 smfresal 46808 smfres 46810 smfco 46822 smffmptf 46824 smfsuplem1 46831 smfsuplem3 46833 smfsupxr 46836 smfinflem 46837 smflimsuplem2 46841 smflimsuplem3 46842 smflimsuplem4 46843 smflimsuplem5 46844 smfliminflem 46850 smfpimne 46859 smfpimne2 46860 smfsupdmmbllem 46864 smfinfdmmbllem 46868 | 
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