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Theorem issmf 46743
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 open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
issmf.s (𝜑𝑆 ∈ SAlg)
issmf.d 𝐷 = dom 𝐹
Assertion
Ref Expression
issmf (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 issmf
Dummy variables 𝑏 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issmf.s . . 3 (𝜑𝑆 ∈ SAlg)
2 issmf.d . . 3 𝐷 = dom 𝐹
31, 2issmflem 46742 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑏} ∈ (𝑆t 𝐷))))
4 breq2 5147 . . . . . . . 8 (𝑏 = 𝑎 → ((𝐹𝑦) < 𝑏 ↔ (𝐹𝑦) < 𝑎))
54rabbidv 3444 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) < 𝑏} = {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎})
65eleq1d 2826 . . . . . 6 (𝑏 = 𝑎 → ({𝑦𝐷 ∣ (𝐹𝑦) < 𝑏} ∈ (𝑆t 𝐷) ↔ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷)))
7 fveq2 6906 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
87breq1d 5153 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑦) < 𝑎 ↔ (𝐹𝑥) < 𝑎))
98cbvrabv 3447 . . . . . . . 8 {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎}
109eleq1i 2832 . . . . . . 7 ({𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
1110a1i 11 . . . . . 6 (𝑏 = 𝑎 → ({𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)))
126, 11bitrd 279 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷 ∣ (𝐹𝑦) < 𝑏} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)))
1312cbvralvw 3237 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑏} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
14133anbi3i 1160 . . 3 ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑏} ∈ (𝑆t 𝐷)) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)))
1514a1i 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))))
163, 15bitrd 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  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {crab 3436  wss 3951   cuni 4907   class class class wbr 5143  dom cdm 5685  wf 6557  cfv 6561  (class class class)co 7431  cr 11154   < clt 11295  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-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230
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-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  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-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-ioo 13391  df-ico 13393  df-smblfn 46711
This theorem is referenced by:  smfpreimalt  46746  smff  46747  smfdmss  46748  issmff  46749  issmfd  46750  issmflelem  46759  issmfgtlem  46770  issmfgelem  46784
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