Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfres | Structured version Visualization version GIF version |
Description: The restriction of sigma-measurable function is sigma-measurable. Proposition 121E (h) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfres.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfres.f | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
smfres.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
smfres | ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1906 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | smfres.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | inss1 4202 | . . . 4 ⊢ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹 | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹) |
5 | smfres.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | |
6 | eqid 2818 | . . . 4 ⊢ dom 𝐹 = dom 𝐹 | |
7 | 2, 5, 6 | smfdmss 42887 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
8 | 4, 7 | sstrd 3974 | . 2 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) ⊆ ∪ 𝑆) |
9 | 2, 5, 6 | smff 42886 | . . 3 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
10 | fresin 6540 | . . 3 ⊢ (𝐹:dom 𝐹⟶ℝ → (𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ) |
12 | ovexd 7180 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t dom 𝐹) ∈ V) | |
13 | smfres.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
14 | 13 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ∈ 𝑉) |
15 | 2 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
16 | 5 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆)) |
17 | mnfxr 10686 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → -∞ ∈ ℝ*) |
19 | rexr 10675 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
20 | 19 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
21 | 15, 16, 6, 18, 20 | smfpimioo 42939 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
22 | eqid 2818 | . . . 4 ⊢ ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) | |
23 | 12, 14, 21, 22 | elrestd 41251 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) ∈ ((𝑆 ↾t dom 𝐹) ↾t 𝐴)) |
24 | 9 | ffund 6511 | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) |
25 | respreima 6828 | . . . . . . . 8 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎)) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴)) | |
26 | 24, 25 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎)) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴)) |
27 | 26 | eqcomd 2824 | . . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎))) |
28 | 27 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎))) |
29 | 11 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ) |
30 | 29, 20 | preimaioomnf 42874 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎)) = {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎}) |
31 | 28, 30 | eqtr2d 2854 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎} = ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴)) |
32 | 5 | dmexd 7604 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 ∈ V) |
33 | restco 21700 | . . . . . . 7 ⊢ ((𝑆 ∈ SAlg ∧ dom 𝐹 ∈ V ∧ 𝐴 ∈ 𝑉) → ((𝑆 ↾t dom 𝐹) ↾t 𝐴) = (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) | |
34 | 2, 32, 13, 33 | syl3anc 1363 | . . . . . 6 ⊢ (𝜑 → ((𝑆 ↾t dom 𝐹) ↾t 𝐴) = (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
35 | 34 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((𝑆 ↾t dom 𝐹) ↾t 𝐴) = (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
36 | 35 | eqcomd 2824 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t (dom 𝐹 ∩ 𝐴)) = ((𝑆 ↾t dom 𝐹) ↾t 𝐴)) |
37 | 31, 36 | eleq12d 2904 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ({𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎} ∈ (𝑆 ↾t (dom 𝐹 ∩ 𝐴)) ↔ ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) ∈ ((𝑆 ↾t dom 𝐹) ↾t 𝐴))) |
38 | 23, 37 | mpbird 258 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎} ∈ (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
39 | 1, 2, 8, 11, 38 | issmfd 42889 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 ∪ cuni 4830 class class class wbr 5057 ◡ccnv 5547 dom cdm 5548 ↾ cres 5550 “ cima 5551 Fun wfun 6342 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 -∞cmnf 10661 ℝ*cxr 10662 < clt 10663 (,)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: (None) |
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