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Theorem issmfle 46760
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 480 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3 simpr 484 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4 issmfle.d . . . . . 6 𝐷 = dom 𝐹
52, 3, 4smfdmss 46748 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 46747 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7 nfv 1914 . . . . . . 7 𝑏𝜑
8 nfv 1914 . . . . . . 7 𝑏 𝐹 ∈ (SMblFn‘𝑆)
97, 8nfan 1899 . . . . . 6 𝑏(𝜑𝐹 ∈ (SMblFn‘𝑆))
10 nfv 1914 . . . . . . . . . 10 𝑦𝜑
11 nfv 1914 . . . . . . . . . 10 𝑦 𝐹 ∈ (SMblFn‘𝑆)
1210, 11nfan 1899 . . . . . . . . 9 𝑦(𝜑𝐹 ∈ (SMblFn‘𝑆))
13 nfv 1914 . . . . . . . . 9 𝑦 𝑏 ∈ ℝ
1412, 13nfan 1899 . . . . . . . 8 𝑦((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
15 nfv 1914 . . . . . . . 8 𝑐((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
161uniexd 7762 . . . . . . . . . . . . 13 (𝜑 𝑆 ∈ V)
1716adantr 480 . . . . . . . . . . . 12 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
18 simpr 484 . . . . . . . . . . . 12 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
1917, 18ssexd 5324 . . . . . . . . . . 11 ((𝜑𝐷 𝑆) → 𝐷 ∈ V)
205, 19syldan 591 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
21 eqid 2737 . . . . . . . . . 10 (𝑆t 𝐷) = (𝑆t 𝐷)
222, 20, 21subsalsal 46374 . . . . . . . . 9 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2322adantr 480 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
246frexr 45396 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ*)
2524adantr 480 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝐹:𝐷⟶ℝ*)
2625ffvelcdmda 7104 . . . . . . . 8 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ*)
272adantr 480 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg)
283adantr 480 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
29 simpr 484 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
3027, 28, 4, 29smfpreimalt 46746 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) < 𝑐} ∈ (𝑆t 𝐷))
3130adantlr 715 . . . . . . . 8 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) < 𝑐} ∈ (𝑆t 𝐷))
32 simpr 484 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
3314, 15, 23, 26, 31, 32salpreimaltle 46741 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
3433ex 412 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
359, 34ralrimi 3257 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
365, 6, 353jca 1129 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
3736ex 412 . . 3 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))))
38 nfv 1914 . . . . . . 7 𝑦 𝐷 𝑆
39 nfv 1914 . . . . . . 7 𝑦 𝐹:𝐷⟶ℝ
40 nfcv 2905 . . . . . . . 8 𝑦
41 nfrab1 3457 . . . . . . . . 9 𝑦{𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏}
42 nfcv 2905 . . . . . . . . 9 𝑦(𝑆t 𝐷)
4341, 42nfel 2920 . . . . . . . 8 𝑦{𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)
4440, 43nfralw 3311 . . . . . . 7 𝑦𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)
4538, 39, 44nf3an 1901 . . . . . 6 𝑦(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
4610, 45nfan 1899 . . . . 5 𝑦(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
47 nfv 1914 . . . . . . 7 𝑏 𝐷 𝑆
48 nfv 1914 . . . . . . 7 𝑏 𝐹:𝐷⟶ℝ
49 nfra1 3284 . . . . . . 7 𝑏𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)
5047, 48, 49nf3an 1901 . . . . . 6 𝑏(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
517, 50nfan 1899 . . . . 5 𝑏(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)))
521adantr 480 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝑆 ∈ SAlg)
53 simpr1 1195 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝐷 𝑆)
54 simpr2 1196 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝐹:𝐷⟶ℝ)
55 rspa 3248 . . . . . . 7 ((∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
56553ad2antl3 1188 . . . . . 6 (((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))
5756adantll 714 . . . . 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 46759 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
5958ex 412 . . 3 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
6037, 59impbid 212 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷))))
61 breq2 5147 . . . . . . . 8 (𝑏 = 𝑎 → ((𝐹𝑦) ≤ 𝑏 ↔ (𝐹𝑦) ≤ 𝑎))
6261rabbidv 3444 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} = {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑎})
63 fveq2 6906 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
6463breq1d 5153 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑦) ≤ 𝑎 ↔ (𝐹𝑥) ≤ 𝑎))
6564cbvrabv 3447 . . . . . . . 8 {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎}
6665a1i 11 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎})
6762, 66eqtrd 2777 . . . . . 6 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} = {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎})
6867eleq1d 2826 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷)))
6968cbvralvw 3237 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑏} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))
70693anbi3i 1160 . . 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 279 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951   cuni 4907   class class class wbr 5143  dom cdm 5685  wf 6557  cfv 6561  (class class class)co 7431  cr 11154  *cxr 11294   < clt 11295  cle 11296  t crest 17465  SAlgcsalg 46323  SMblFncsmblfn 46710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-card 9979  df-acn 9982  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-ioo 13391  df-ico 13393  df-fl 13832  df-rest 17467  df-salg 46324  df-smblfn 46711
This theorem is referenced by:  smfpreimale  46769  issmfgt  46771  issmfled  46772
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