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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 41733 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
5 | 4 | feqmptd 6500 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥))) |
6 | 5 | cnveqd 5534 | . . . 4 ⊢ (𝜑 → ◡𝐹 = ◡(𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥))) |
7 | 6 | imaeq1d 5710 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) = (◡(𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) “ (𝐴(,)𝐵))) |
8 | eqid 2825 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) | |
9 | 8 | mptpreima 5873 | . . . 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 2861 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ (𝐴(,)𝐵)}) |
12 | nfv 2013 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
13 | 1 | uniexd 40097 | . . . 4 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
14 | 1, 2, 3 | smfdmss 41734 | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
15 | 13, 14 | ssexd 5032 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
16 | 4 | ffvelrnda 6613 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ ℝ) |
17 | 5, 2 | eqeltrrd 2907 | . . 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 41785 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ (𝐴(,)𝐵)} ∈ (𝑆 ↾t 𝐷)) |
21 | 11, 20 | eqeltrd 2906 | 1 ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) ∈ (𝑆 ↾t 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 {crab 3121 Vcvv 3414 ∪ cuni 4660 ↦ cmpt 4954 ◡ccnv 5345 dom cdm 5346 “ cima 5349 ‘cfv 6127 (class class class)co 6910 ℝcr 10258 ℝ*cxr 10397 (,)cioo 12470 ↾t crest 16441 SAlgcsalg 41317 SMblFncsmblfn 41701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cc 9579 ax-ac2 9607 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-inf 8624 df-card 9085 df-acn 9088 df-ac 9259 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-n0 11626 df-z 11712 df-uz 11976 df-q 12079 df-rp 12120 df-ioo 12474 df-ico 12476 df-fl 12895 df-rest 16443 df-salg 41318 df-smblfn 41702 |
This theorem is referenced by: smfres 41789 smfpimbor1lem1 41797 |
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