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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 1873 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | smfres.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | inss1 4092 | . . . 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 2778 | . . . 4 ⊢ dom 𝐹 = dom 𝐹 | |
7 | 2, 5, 6 | smfdmss 42447 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
8 | 4, 7 | sstrd 3868 | . 2 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) ⊆ ∪ 𝑆) |
9 | 2, 5, 6 | smff 42446 | . . 3 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
10 | fresin 6376 | . . 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 7010 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t dom 𝐹) ∈ V) | |
13 | smfres.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
14 | 13 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ∈ 𝑉) |
15 | 2 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
16 | 5 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆)) |
17 | mnfxr 10498 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → -∞ ∈ ℝ*) |
19 | rexr 10486 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
20 | 19 | adantl 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
21 | 15, 16, 6, 18, 20 | smfpimioo 42499 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
22 | eqid 2778 | . . . 4 ⊢ ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) | |
23 | 12, 14, 21, 22 | elrestd 40803 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) ∈ ((𝑆 ↾t dom 𝐹) ↾t 𝐴)) |
24 | 9 | ffund 6348 | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) |
25 | respreima 6661 | . . . . . . . 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 2784 | . . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎))) |
28 | 27 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎))) |
29 | 11 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ) |
30 | 29, 20 | preimaioomnf 42434 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎)) = {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎}) |
31 | 28, 30 | eqtr2d 2815 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎} = ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴)) |
32 | 5 | dmexd 7430 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 ∈ V) |
33 | restco 21476 | . . . . . . 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 1351 | . . . . . 6 ⊢ (𝜑 → ((𝑆 ↾t dom 𝐹) ↾t 𝐴) = (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
35 | 34 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((𝑆 ↾t dom 𝐹) ↾t 𝐴) = (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
36 | 35 | eqcomd 2784 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t (dom 𝐹 ∩ 𝐴)) = ((𝑆 ↾t dom 𝐹) ↾t 𝐴)) |
37 | 31, 36 | eleq12d 2860 | . . 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 249 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎} ∈ (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
39 | 1, 2, 8, 11, 38 | issmfd 42449 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 {crab 3092 Vcvv 3415 ∩ cin 3828 ⊆ wss 3829 ∪ cuni 4712 class class class wbr 4929 ◡ccnv 5406 dom cdm 5407 ↾ cres 5409 “ cima 5410 Fun wfun 6182 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ℝcr 10334 -∞cmnf 10472 ℝ*cxr 10473 < clt 10474 (,)cioo 12554 ↾t crest 16550 SAlgcsalg 42030 SMblFncsmblfn 42414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cc 9655 ax-ac2 9683 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-inf 8702 df-card 9162 df-acn 9165 df-ac 9336 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-n0 11708 df-z 11794 df-uz 12059 df-q 12163 df-rp 12205 df-ioo 12558 df-ico 12560 df-fl 12977 df-rest 16552 df-salg 42031 df-smblfn 42415 |
This theorem is referenced by: (None) |
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