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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfpimioo | Structured version Visualization version GIF version |
Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfpimioo.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfpimioo.f | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
smfpimioo.d | ⊢ 𝐷 = dom 𝐹 |
smfpimioo.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
smfpimioo.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
smfpimioo | ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) ∈ (𝑆 ↾t 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfpimioo.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
2 | smfpimioo.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | |
3 | smfpimioo.d | . . . . . . 7 ⊢ 𝐷 = dom 𝐹 | |
4 | 1, 2, 3 | smff 41861 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
5 | 4 | feqmptd 6509 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥))) |
6 | 5 | cnveqd 5543 | . . . 4 ⊢ (𝜑 → ◡𝐹 = ◡(𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥))) |
7 | 6 | imaeq1d 5719 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) = (◡(𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) “ (𝐴(,)𝐵))) |
8 | eqid 2777 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) | |
9 | 8 | mptpreima 5882 | . . . 4 ⊢ (◡(𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) “ (𝐴(,)𝐵)) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ (𝐴(,)𝐵)} |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) “ (𝐴(,)𝐵)) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ (𝐴(,)𝐵)}) |
11 | 7, 10 | eqtrd 2813 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ (𝐴(,)𝐵)}) |
12 | nfv 1957 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
13 | 1 | uniexd 40205 | . . . 4 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
14 | 1, 2, 3 | smfdmss 41862 | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
15 | 13, 14 | ssexd 5042 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
16 | 4 | ffvelrnda 6623 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ ℝ) |
17 | 5, 2 | eqeltrrd 2859 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) ∈ (SMblFn‘𝑆)) |
18 | smfpimioo.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
19 | smfpimioo.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
20 | 12, 1, 15, 16, 17, 18, 19 | smfpimioompt 41913 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ (𝐴(,)𝐵)} ∈ (𝑆 ↾t 𝐷)) |
21 | 11, 20 | eqeltrd 2858 | 1 ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) ∈ (𝑆 ↾t 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 {crab 3093 Vcvv 3397 ∪ cuni 4671 ↦ cmpt 4965 ◡ccnv 5354 dom cdm 5355 “ cima 5358 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 ℝ*cxr 10410 (,)cioo 12487 ↾t crest 16467 SAlgcsalg 41445 SMblFncsmblfn 41829 |
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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cc 9592 ax-ac2 9620 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
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 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 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-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 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-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-card 9098 df-acn 9101 df-ac 9272 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 df-q 12096 df-rp 12138 df-ioo 12491 df-ico 12493 df-fl 12912 df-rest 16469 df-salg 41446 df-smblfn 41830 |
This theorem is referenced by: smfres 41917 smfpimbor1lem1 41925 |
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