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Theorem smfpreimaltf 46751
Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
smfpreimaltf.x 𝑥𝐹
smfpreimaltf.s (𝜑𝑆 ∈ SAlg)
smfpreimaltf.f (𝜑𝐹 ∈ (SMblFn‘𝑆))
smfpreimaltf.d 𝐷 = dom 𝐹
smfpreimaltf.a (𝜑𝐴 ∈ ℝ)
Assertion
Ref Expression
smfpreimaltf (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐷(𝑥)   𝑆(𝑥)   𝐹(𝑥)

Proof of Theorem smfpreimaltf
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 smfpreimaltf.a . 2 (𝜑𝐴 ∈ ℝ)
2 smfpreimaltf.f . . . 4 (𝜑𝐹 ∈ (SMblFn‘𝑆))
3 smfpreimaltf.x . . . . 5 𝑥𝐹
4 smfpreimaltf.s . . . . 5 (𝜑𝑆 ∈ SAlg)
5 smfpreimaltf.d . . . . 5 𝐷 = dom 𝐹
63, 4, 5issmff 46749 . . . 4 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
72, 6mpbid 232 . . 3 (𝜑 → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)))
87simp3d 1145 . 2 (𝜑 → ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
9 breq2 5147 . . . . 5 (𝑎 = 𝐴 → ((𝐹𝑥) < 𝑎 ↔ (𝐹𝑥) < 𝐴))
109rabbidv 3444 . . . 4 (𝑎 = 𝐴 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴})
1110eleq1d 2826 . . 3 (𝑎 = 𝐴 → ({𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷)))
1211rspcva 3620 . 2 ((𝐴 ∈ ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))
131, 8, 12syl2anc 584 1 (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  wnfc 2890  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:  smfpimltmpt  46761  smfpimltxr  46762
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