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Theorem issmfle 40258
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 481 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3 simpr 477 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4 issmfle.d . . . . . 6 𝐷 = dom 𝐹
52, 3, 4smfdmss 40246 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 40245 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7 nfv 1840 . . . . . . 7 𝑏𝜑
8 nfv 1840 . . . . . . 7 𝑏 𝐹 ∈ (SMblFn‘𝑆)
97, 8nfan 1825 . . . . . 6 𝑏(𝜑𝐹 ∈ (SMblFn‘𝑆))
10 nfv 1840 . . . . . . . . . 10 𝑦𝜑
11 nfv 1840 . . . . . . . . . 10 𝑦 𝐹 ∈ (SMblFn‘𝑆)
1210, 11nfan 1825 . . . . . . . . 9 𝑦(𝜑𝐹 ∈ (SMblFn‘𝑆))
13 nfv 1840 . . . . . . . . 9 𝑦 𝑏 ∈ ℝ
1412, 13nfan 1825 . . . . . . . 8 𝑦((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
15 nfv 1840 . . . . . . . 8 𝑐((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
161uniexd 38763 . . . . . . . . . . . . 13 (𝜑 𝑆 ∈ V)
1716adantr 481 . . . . . . . . . . . 12 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
18 simpr 477 . . . . . . . . . . . 12 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
1917, 18ssexd 4765 . . . . . . . . . . 11 ((𝜑𝐷 𝑆) → 𝐷 ∈ V)
205, 19syldan 487 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
21 eqid 2621 . . . . . . . . . 10 (𝑆t 𝐷) = (𝑆t 𝐷)
222, 20, 21subsalsal 39881 . . . . . . . . 9 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2322adantr 481 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
246frexr 39065 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ*)
2524adantr 481 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝐹:𝐷⟶ℝ*)
2625ffvelrnda 6315 . . . . . . . 8 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ*)
272adantr 481 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg)
283adantr 481 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
29 simpr 477 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
3027, 28, 4, 29smfpreimalt 40244 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) < 𝑐} ∈ (𝑆t 𝐷))
3130adantlr 750 . . . . . . . 8 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) < 𝑐} ∈ (𝑆t 𝐷))
32 simpr 477 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
3314, 15, 23, 26, 31, 32salpreimaltle 40239 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
3433ex 450 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
359, 34ralrimi 2951 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
365, 6, 353jca 1240 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
3736ex 450 . . 3 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))))
38 nfv 1840 . . . . . . 7 𝑦 𝐷 𝑆
39 nfv 1840 . . . . . . 7 𝑦 𝐹:𝐷⟶ℝ
40 nfcv 2761 . . . . . . . 8 𝑦
41 nfrab1 3111 . . . . . . . . 9 𝑦{𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏}
42 nfcv 2761 . . . . . . . . 9 𝑦(𝑆t 𝐷)
4341, 42nfel 2773 . . . . . . . 8 𝑦{𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)
4440, 43nfral 2940 . . . . . . 7 𝑦𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)
4538, 39, 44nf3an 1828 . . . . . 6 𝑦(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
4610, 45nfan 1825 . . . . 5 𝑦(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
47 nfv 1840 . . . . . . 7 𝑏 𝐷 𝑆
48 nfv 1840 . . . . . . 7 𝑏 𝐹:𝐷⟶ℝ
49 nfra1 2936 . . . . . . 7 𝑏𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)
5047, 48, 49nf3an 1828 . . . . . 6 𝑏(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
517, 50nfan 1825 . . . . 5 𝑏(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
521adantr 481 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝑆 ∈ SAlg)
53 simpr1 1065 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝐷 𝑆)
54 simpr2 1066 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝐹:𝐷⟶ℝ)
55 rspa 2925 . . . . . . 7 ((∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
56553ad2antl3 1223 . . . . . 6 (((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
5756adantll 749 . . . . 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 40257 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
5958ex 450 . . 3 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
6037, 59impbid 202 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))))
61 breq2 4617 . . . . . . . 8 (𝑏 = 𝑎 → ((𝐹𝑦) ≤ 𝑏 ↔ (𝐹𝑦) ≤ 𝑎))
6261rabbidv 3177 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} = {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑎})
63 fveq2 6148 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
6463breq1d 4623 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑦) ≤ 𝑎 ↔ (𝐹𝑥) ≤ 𝑎))
6564cbvrabv 3185 . . . . . . . 8 {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎}
6665a1i 11 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎})
6762, 66eqtrd 2655 . . . . . 6 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} = {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎})
6867eleq1d 2683 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷)))
6968cbvralv 3159 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))
70693anbi3i 1253 . . 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 268 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  wss 3555   cuni 4402   class class class wbr 4613  dom cdm 5074  wf 5843  cfv 5847  (class class class)co 6604  cr 9879  *cxr 10017   < clt 10018  cle 10019  t crest 16002  SAlgcsalg 39832  SMblFncsmblfn 40213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cc 9201  ax-ac2 9229  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-card 8709  df-acn 8712  df-ac 8883  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-ioo 12121  df-ico 12123  df-fl 12533  df-rest 16004  df-salg 39833  df-smblfn 40214
This theorem is referenced by:  smfpreimale  40267  issmfgt  40269  issmfled  40270
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