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Theorem issmfle 43376
Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all right-closed intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (ii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
issmfle.s (𝜑𝑆 ∈ SAlg)
issmfle.d 𝐷 = dom 𝐹
Assertion
Ref Expression
issmfle (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))))
Distinct variable groups:   𝐷,𝑎,𝑥   𝐹,𝑎,𝑥   𝑆,𝑎
Allowed substitution hints:   𝜑(𝑥,𝑎)   𝑆(𝑥)

Proof of Theorem issmfle
Dummy variables 𝑏 𝑦 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issmfle.s . . . . . . 7 (𝜑𝑆 ∈ SAlg)
21adantr 484 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3 simpr 488 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4 issmfle.d . . . . . 6 𝐷 = dom 𝐹
52, 3, 4smfdmss 43364 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 43363 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7 nfv 1915 . . . . . . 7 𝑏𝜑
8 nfv 1915 . . . . . . 7 𝑏 𝐹 ∈ (SMblFn‘𝑆)
97, 8nfan 1900 . . . . . 6 𝑏(𝜑𝐹 ∈ (SMblFn‘𝑆))
10 nfv 1915 . . . . . . . . . 10 𝑦𝜑
11 nfv 1915 . . . . . . . . . 10 𝑦 𝐹 ∈ (SMblFn‘𝑆)
1210, 11nfan 1900 . . . . . . . . 9 𝑦(𝜑𝐹 ∈ (SMblFn‘𝑆))
13 nfv 1915 . . . . . . . . 9 𝑦 𝑏 ∈ ℝ
1412, 13nfan 1900 . . . . . . . 8 𝑦((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
15 nfv 1915 . . . . . . . 8 𝑐((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
161uniexd 7452 . . . . . . . . . . . . 13 (𝜑 𝑆 ∈ V)
1716adantr 484 . . . . . . . . . . . 12 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
18 simpr 488 . . . . . . . . . . . 12 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
1917, 18ssexd 5195 . . . . . . . . . . 11 ((𝜑𝐷 𝑆) → 𝐷 ∈ V)
205, 19syldan 594 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
21 eqid 2801 . . . . . . . . . 10 (𝑆t 𝐷) = (𝑆t 𝐷)
222, 20, 21subsalsal 42996 . . . . . . . . 9 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2322adantr 484 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
246frexr 42016 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ*)
2524adantr 484 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝐹:𝐷⟶ℝ*)
2625ffvelrnda 6832 . . . . . . . 8 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ*)
272adantr 484 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg)
283adantr 484 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
29 simpr 488 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
3027, 28, 4, 29smfpreimalt 43362 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) < 𝑐} ∈ (𝑆t 𝐷))
3130adantlr 714 . . . . . . . 8 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) < 𝑐} ∈ (𝑆t 𝐷))
32 simpr 488 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
3314, 15, 23, 26, 31, 32salpreimaltle 43357 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
3433ex 416 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
359, 34ralrimi 3183 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
365, 6, 353jca 1125 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
3736ex 416 . . 3 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))))
38 nfv 1915 . . . . . . 7 𝑦 𝐷 𝑆
39 nfv 1915 . . . . . . 7 𝑦 𝐹:𝐷⟶ℝ
40 nfcv 2958 . . . . . . . 8 𝑦
41 nfrab1 3340 . . . . . . . . 9 𝑦{𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏}
42 nfcv 2958 . . . . . . . . 9 𝑦(𝑆t 𝐷)
4341, 42nfel 2972 . . . . . . . 8 𝑦{𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)
4440, 43nfralw 3192 . . . . . . 7 𝑦𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)
4538, 39, 44nf3an 1902 . . . . . 6 𝑦(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
4610, 45nfan 1900 . . . . 5 𝑦(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
47 nfv 1915 . . . . . . 7 𝑏 𝐷 𝑆
48 nfv 1915 . . . . . . 7 𝑏 𝐹:𝐷⟶ℝ
49 nfra1 3186 . . . . . . 7 𝑏𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)
5047, 48, 49nf3an 1902 . . . . . 6 𝑏(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
517, 50nfan 1900 . . . . 5 𝑏(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
521adantr 484 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝑆 ∈ SAlg)
53 simpr1 1191 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝐷 𝑆)
54 simpr2 1192 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝐹:𝐷⟶ℝ)
55 rspa 3174 . . . . . . 7 ((∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
56553ad2antl3 1184 . . . . . 6 (((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
5756adantll 713 . . . . 5 (((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
5846, 51, 52, 4, 53, 54, 57issmflelem 43375 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
5958ex 416 . . 3 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
6037, 59impbid 215 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))))
61 breq2 5037 . . . . . . . 8 (𝑏 = 𝑎 → ((𝐹𝑦) ≤ 𝑏 ↔ (𝐹𝑦) ≤ 𝑎))
6261rabbidv 3430 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} = {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑎})
63 fveq2 6649 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
6463breq1d 5043 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑦) ≤ 𝑎 ↔ (𝐹𝑥) ≤ 𝑎))
6564cbvrabv 3442 . . . . . . . 8 {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎}
6665a1i 11 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎})
6762, 66eqtrd 2836 . . . . . 6 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} = {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎})
6867eleq1d 2877 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷)))
6968cbvralvw 3399 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))
70693anbi3i 1156 . . 3 ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷)))
7170a1i 11 . 2 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))))
7260, 71bitrd 282 1 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  {crab 3113  Vcvv 3444  wss 3884   cuni 4803   class class class wbr 5033  dom cdm 5523  wf 6324  cfv 6328  (class class class)co 7139  cr 10529  *cxr 10667   < clt 10668  cle 10669  t crest 16690  SAlgcsalg 42947  SMblFncsmblfn 43331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cc 9850  ax-ac2 9878  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-card 9356  df-acn 9359  df-ac 9531  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-ioo 12734  df-ico 12736  df-fl 13161  df-rest 16692  df-salg 42948  df-smblfn 43332
This theorem is referenced by:  smfpreimale  43385  issmfgt  43387  issmfled  43388
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