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 1843 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | smfres.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | inss1 3833 | . . . 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 2622 | . . . 4 ⊢ dom 𝐹 = dom 𝐹 | |
7 | 2, 5, 6 | smfdmss 40942 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
8 | 4, 7 | sstrd 3613 | . 2 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) ⊆ ∪ 𝑆) |
9 | 2, 5, 6 | smff 40941 | . . 3 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
10 | fresin 6073 | . . 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 6680 | . . . 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 10096 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → -∞ ∈ ℝ*) |
19 | rexr 10085 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
20 | 19 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
21 | 15, 16, 6, 18, 20 | smfpimioo 40994 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
22 | eqid 2622 | . . . 4 ⊢ ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) | |
23 | 12, 14, 21, 22 | elrestd 39291 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) ∈ ((𝑆 ↾t dom 𝐹) ↾t 𝐴)) |
24 | 9 | ffund 6049 | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) |
25 | respreima 6344 | . . . . . . . 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 2628 | . . . . . 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 40929 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎)) = {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎}) |
31 | 28, 30 | eqtr2d 2657 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎} = ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴)) |
32 | 5 | dmexd 39422 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 ∈ V) |
33 | restco 20968 | . . . . . . 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 1326 | . . . . . 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 2628 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t (dom 𝐹 ∩ 𝐴)) = ((𝑆 ↾t dom 𝐹) ↾t 𝐴)) |
37 | 31, 36 | eleq12d 2695 | . . 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 247 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎} ∈ (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
39 | 1, 2, 8, 11, 38 | issmfd 40944 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ∪ cuni 4436 class class class wbr 4653 ◡ccnv 5113 dom cdm 5114 ↾ cres 5116 “ cima 5117 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 -∞cmnf 10072 ℝ*cxr 10073 < clt 10074 (,)cioo 12175 ↾t crest 16081 SAlgcsalg 40528 SMblFncsmblfn 40909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-ac2 9285 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-acn 8768 df-ac 8939 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-ioo 12179 df-ico 12181 df-fl 12593 df-rest 16083 df-salg 40529 df-smblfn 40910 |
This theorem is referenced by: (None) |
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