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 1915 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | smfres.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | inss1 4207 | . . . 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 2823 | . . . 4 ⊢ dom 𝐹 = dom 𝐹 | |
7 | 2, 5, 6 | smfdmss 43017 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
8 | 4, 7 | sstrd 3979 | . 2 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) ⊆ ∪ 𝑆) |
9 | 2, 5, 6 | smff 43016 | . . 3 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
10 | fresin 6549 | . . 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 7193 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t dom 𝐹) ∈ V) | |
13 | smfres.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
14 | 13 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ∈ 𝑉) |
15 | 2 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
16 | 5 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆)) |
17 | mnfxr 10700 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → -∞ ∈ ℝ*) |
19 | rexr 10689 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
20 | 19 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
21 | 15, 16, 6, 18, 20 | smfpimioo 43069 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
22 | eqid 2823 | . . . 4 ⊢ ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) | |
23 | 12, 14, 21, 22 | elrestd 41381 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) ∈ ((𝑆 ↾t dom 𝐹) ↾t 𝐴)) |
24 | 9 | ffund 6520 | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) |
25 | respreima 6838 | . . . . . . . 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 2829 | . . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎))) |
28 | 27 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎))) |
29 | 11 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ) |
30 | 29, 20 | preimaioomnf 43004 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎)) = {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎}) |
31 | 28, 30 | eqtr2d 2859 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎} = ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴)) |
32 | 5 | dmexd 7617 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 ∈ V) |
33 | restco 21774 | . . . . . . 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 1367 | . . . . . 6 ⊢ (𝜑 → ((𝑆 ↾t dom 𝐹) ↾t 𝐴) = (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
35 | 34 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((𝑆 ↾t dom 𝐹) ↾t 𝐴) = (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
36 | 35 | eqcomd 2829 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t (dom 𝐹 ∩ 𝐴)) = ((𝑆 ↾t dom 𝐹) ↾t 𝐴)) |
37 | 31, 36 | eleq12d 2909 | . . 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 259 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎} ∈ (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
39 | 1, 2, 8, 11, 38 | issmfd 43019 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 ∪ cuni 4840 class class class wbr 5068 ◡ccnv 5556 dom cdm 5557 ↾ cres 5559 “ cima 5560 Fun wfun 6351 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 -∞cmnf 10675 ℝ*cxr 10676 < clt 10677 (,)cioo 12741 ↾t crest 16696 SAlgcsalg 42600 SMblFncsmblfn 42984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-ac2 9887 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-card 9370 df-acn 9373 df-ac 9544 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-ioo 12745 df-ico 12747 df-fl 13165 df-rest 16698 df-salg 42601 df-smblfn 42985 |
This theorem is referenced by: (None) |
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