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 43003 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
5 | 4 | feqmptd 6727 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥))) |
6 | 5 | cnveqd 5740 | . . . 4 ⊢ (𝜑 → ◡𝐹 = ◡(𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥))) |
7 | 6 | imaeq1d 5922 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) = (◡(𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) “ (𝐴(,)𝐵))) |
8 | eqid 2821 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) | |
9 | 8 | mptpreima 6086 | . . . 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 2856 | . 2 ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ (𝐴(,)𝐵)}) |
12 | nfv 1911 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
13 | 1 | uniexd 7462 | . . . 4 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
14 | 1, 2, 3 | smfdmss 43004 | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
15 | 13, 14 | ssexd 5220 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
16 | 4 | ffvelrnda 6845 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ ℝ) |
17 | 5, 2 | eqeltrrd 2914 | . . 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 43055 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ∈ (𝐴(,)𝐵)} ∈ (𝑆 ↾t 𝐷)) |
21 | 11, 20 | eqeltrd 2913 | 1 ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) ∈ (𝑆 ↾t 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ∪ cuni 4831 ↦ cmpt 5138 ◡ccnv 5548 dom cdm 5549 “ cima 5552 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 ℝ*cxr 10668 (,)cioo 12732 ↾t crest 16688 SAlgcsalg 42587 SMblFncsmblfn 42971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cc 9851 ax-ac2 9879 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-card 9362 df-acn 9365 df-ac 9536 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-ioo 12736 df-ico 12738 df-fl 13156 df-rest 16690 df-salg 42588 df-smblfn 42972 |
This theorem is referenced by: smfres 43059 smfpimbor1lem1 43067 |
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