![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 1910 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | smfres.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | inss1 4225 | . . . 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 2728 | . . . 4 ⊢ dom 𝐹 = dom 𝐹 | |
7 | 2, 5, 6 | smfdmss 46112 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
8 | 4, 7 | sstrd 3989 | . 2 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) ⊆ ∪ 𝑆) |
9 | 2, 5, 6 | smff 46111 | . . 3 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
10 | fresin 6761 | . . 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 7450 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t dom 𝐹) ∈ V) | |
13 | smfres.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
14 | 13 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ∈ 𝑉) |
15 | 2 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
16 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆)) |
17 | mnfxr 11296 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → -∞ ∈ ℝ*) |
19 | rexr 11285 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
20 | 19 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
21 | 15, 16, 6, 18, 20 | smfpimioo 46166 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
22 | eqid 2728 | . . . 4 ⊢ ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) | |
23 | 12, 14, 21, 22 | elrestd 44465 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) ∈ ((𝑆 ↾t dom 𝐹) ↾t 𝐴)) |
24 | 9 | ffund 6721 | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) |
25 | respreima 7070 | . . . . . . . 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 2734 | . . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎))) |
28 | 27 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴) = (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎))) |
29 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ) |
30 | 29, 20 | preimaioomnf 46098 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡(𝐹 ↾ 𝐴) “ (-∞(,)𝑎)) = {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎}) |
31 | 28, 30 | eqtr2d 2769 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎} = ((◡𝐹 “ (-∞(,)𝑎)) ∩ 𝐴)) |
32 | 5 | dmexd 7906 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 ∈ V) |
33 | restco 23062 | . . . . . . 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 1369 | . . . . . 6 ⊢ (𝜑 → ((𝑆 ↾t dom 𝐹) ↾t 𝐴) = (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
35 | 34 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ((𝑆 ↾t dom 𝐹) ↾t 𝐴) = (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
36 | 35 | eqcomd 2734 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t (dom 𝐹 ∩ 𝐴)) = ((𝑆 ↾t dom 𝐹) ↾t 𝐴)) |
37 | 31, 36 | eleq12d 2823 | . . 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 257 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (dom 𝐹 ∩ 𝐴) ∣ ((𝐹 ↾ 𝐴)‘𝑥) < 𝑎} ∈ (𝑆 ↾t (dom 𝐹 ∩ 𝐴))) |
39 | 1, 2, 8, 11, 38 | issmfd 46114 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3428 Vcvv 3470 ∩ cin 3944 ⊆ wss 3945 ∪ cuni 4904 class class class wbr 5143 ◡ccnv 5672 dom cdm 5673 ↾ cres 5675 “ cima 5676 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 (class class class)co 7415 ℝcr 11132 -∞cmnf 11271 ℝ*cxr 11272 < clt 11273 (,)cioo 13351 ↾t crest 17396 SAlgcsalg 45687 SMblFncsmblfn 46074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cc 10453 ax-ac2 10481 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-pm 8842 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-sup 9460 df-inf 9461 df-card 9957 df-acn 9960 df-ac 10134 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-q 12958 df-rp 13002 df-ioo 13355 df-ico 13357 df-fl 13784 df-rest 17398 df-salg 45688 df-smblfn 46075 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |