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Theorem issmfle 41469
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 466 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3 simpr 471 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4 issmfle.d . . . . . 6 𝐷 = dom 𝐹
52, 3, 4smfdmss 41457 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 41456 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7 nfv 1995 . . . . . . 7 𝑏𝜑
8 nfv 1995 . . . . . . 7 𝑏 𝐹 ∈ (SMblFn‘𝑆)
97, 8nfan 1980 . . . . . 6 𝑏(𝜑𝐹 ∈ (SMblFn‘𝑆))
10 nfv 1995 . . . . . . . . . 10 𝑦𝜑
11 nfv 1995 . . . . . . . . . 10 𝑦 𝐹 ∈ (SMblFn‘𝑆)
1210, 11nfan 1980 . . . . . . . . 9 𝑦(𝜑𝐹 ∈ (SMblFn‘𝑆))
13 nfv 1995 . . . . . . . . 9 𝑦 𝑏 ∈ ℝ
1412, 13nfan 1980 . . . . . . . 8 𝑦((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
15 nfv 1995 . . . . . . . 8 𝑐((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
161uniexd 39800 . . . . . . . . . . . . 13 (𝜑 𝑆 ∈ V)
1716adantr 466 . . . . . . . . . . . 12 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
18 simpr 471 . . . . . . . . . . . 12 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
1917, 18ssexd 4940 . . . . . . . . . . 11 ((𝜑𝐷 𝑆) → 𝐷 ∈ V)
205, 19syldan 579 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
21 eqid 2771 . . . . . . . . . 10 (𝑆t 𝐷) = (𝑆t 𝐷)
222, 20, 21subsalsal 41089 . . . . . . . . 9 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2322adantr 466 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
246frexr 40115 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ*)
2524adantr 466 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝐹:𝐷⟶ℝ*)
2625ffvelrnda 6504 . . . . . . . 8 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ*)
272adantr 466 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg)
283adantr 466 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
29 simpr 471 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
3027, 28, 4, 29smfpreimalt 41455 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) < 𝑐} ∈ (𝑆t 𝐷))
3130adantlr 694 . . . . . . . 8 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) < 𝑐} ∈ (𝑆t 𝐷))
32 simpr 471 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
3314, 15, 23, 26, 31, 32salpreimaltle 41450 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
3433ex 397 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
359, 34ralrimi 3106 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
365, 6, 353jca 1122 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
3736ex 397 . . 3 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))))
38 nfv 1995 . . . . . . 7 𝑦 𝐷 𝑆
39 nfv 1995 . . . . . . 7 𝑦 𝐹:𝐷⟶ℝ
40 nfcv 2913 . . . . . . . 8 𝑦
41 nfrab1 3271 . . . . . . . . 9 𝑦{𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏}
42 nfcv 2913 . . . . . . . . 9 𝑦(𝑆t 𝐷)
4341, 42nfel 2926 . . . . . . . 8 𝑦{𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)
4440, 43nfral 3094 . . . . . . 7 𝑦𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)
4538, 39, 44nf3an 1983 . . . . . 6 𝑦(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
4610, 45nfan 1980 . . . . 5 𝑦(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
47 nfv 1995 . . . . . . 7 𝑏 𝐷 𝑆
48 nfv 1995 . . . . . . 7 𝑏 𝐹:𝐷⟶ℝ
49 nfra1 3090 . . . . . . 7 𝑏𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)
5047, 48, 49nf3an 1983 . . . . . 6 𝑏(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
517, 50nfan 1980 . . . . 5 𝑏(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
521adantr 466 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝑆 ∈ SAlg)
53 simpr1 1233 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝐷 𝑆)
54 simpr2 1235 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝐹:𝐷⟶ℝ)
55 rspa 3079 . . . . . . 7 ((∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
56553ad2antl3 1202 . . . . . 6 (((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
5756adantll 693 . . . . 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 41468 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
5958ex 397 . . 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 4791 . . . . . . . 8 (𝑏 = 𝑎 → ((𝐹𝑦) ≤ 𝑏 ↔ (𝐹𝑦) ≤ 𝑎))
6261rabbidv 3339 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} = {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑎})
63 fveq2 6333 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
6463breq1d 4797 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑦) ≤ 𝑎 ↔ (𝐹𝑥) ≤ 𝑎))
6564cbvrabv 3349 . . . . . . . 8 {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎}
6665a1i 11 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎})
6762, 66eqtrd 2805 . . . . . 6 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} = {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎})
6867eleq1d 2835 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷)))
6968cbvralv 3320 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))
70693anbi3i 1162 . . 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 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  wss 3723   cuni 4575   class class class wbr 4787  dom cdm 5250  wf 6026  cfv 6030  (class class class)co 6796  cr 10141  *cxr 10279   < clt 10280  cle 10281  t crest 16289  SAlgcsalg 41040  SMblFncsmblfn 41424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-inf2 8706  ax-cc 9463  ax-ac2 9491  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8508  df-inf 8509  df-card 8969  df-acn 8972  df-ac 9143  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-n0 11500  df-z 11585  df-uz 11894  df-q 11997  df-rp 12036  df-ioo 12384  df-ico 12386  df-fl 12801  df-rest 16291  df-salg 41041  df-smblfn 41425
This theorem is referenced by:  smfpreimale  41478  issmfgt  41480  issmfled  41481
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