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 42886 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
5 | 4 | feqmptd 6726 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥))) |
6 | 5 | cnveqd 5739 | . . . 4 ⊢ (𝜑 → ◡𝐹 = ◡(𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥))) |
7 | 6 | imaeq1d 5921 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) = (◡(𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) “ (𝐴(,)𝐵))) |
8 | eqid 2818 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) | |
9 | 8 | mptpreima 6085 | . . . 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 2853 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ (𝐴(,)𝐵)}) |
12 | nfv 1906 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
13 | 1 | uniexd 7457 | . . . 4 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
14 | 1, 2, 3 | smfdmss 42887 | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
15 | 13, 14 | ssexd 5219 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
16 | 4 | ffvelrnda 6843 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ ℝ) |
17 | 5, 2 | eqeltrrd 2911 | . . 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 42938 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ (𝐴(,)𝐵)} ∈ (𝑆 ↾t 𝐷)) |
21 | 11, 20 | eqeltrd 2910 | 1 ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) ∈ (𝑆 ↾t 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ∪ cuni 4830 ↦ cmpt 5137 ◡ccnv 5547 dom cdm 5548 “ cima 5551 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 ℝ*cxr 10662 (,)cioo 12726 ↾t crest 16682 SAlgcsalg 42470 SMblFncsmblfn 42854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cc 9845 ax-ac2 9873 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-card 9356 df-acn 9359 df-ac 9530 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-ioo 12730 df-ico 12732 df-fl 13150 df-rest 16684 df-salg 42471 df-smblfn 42855 |
This theorem is referenced by: smfres 42942 smfpimbor1lem1 42950 |
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