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Theorem issmf 46684
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 46683 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑏} ∈ (𝑆t 𝐷))))
4 breq2 5152 . . . . . . . 8 (𝑏 = 𝑎 → ((𝐹𝑦) < 𝑏 ↔ (𝐹𝑦) < 𝑎))
54rabbidv 3441 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷 ∣ (𝐹𝑦) < 𝑏} = {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎})
65eleq1d 2824 . . . . . 6 (𝑏 = 𝑎 → ({𝑦𝐷 ∣ (𝐹𝑦) < 𝑏} ∈ (𝑆t 𝐷) ↔ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷)))
7 fveq2 6907 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
87breq1d 5158 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑦) < 𝑎 ↔ (𝐹𝑥) < 𝑎))
98cbvrabv 3444 . . . . . . . 8 {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎}
109eleq1i 2830 . . . . . . 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 3235 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑏} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
14133anbi3i 1158 . . 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 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433  wss 3963   cuni 4912   class class class wbr 5148  dom cdm 5689  wf 6559  cfv 6563  (class class class)co 7431  cr 11152   < clt 11293  t crest 17467  SAlgcsalg 46264  SMblFncsmblfn 46651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-er 8744  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-ioo 13388  df-ico 13390  df-smblfn 46652
This theorem is referenced by:  smfpreimalt  46687  smff  46688  smfdmss  46689  issmff  46690  issmfd  46691  issmflelem  46700  issmfgtlem  46711  issmfgelem  46725
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