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Theorem smfpreimaltf 40249
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 40247 . . . 4 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
72, 6mpbid 222 . . 3 (𝜑 → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)))
87simp3d 1073 . 2 (𝜑 → ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
9 breq2 4617 . . . . 5 (𝑎 = 𝐴 → ((𝐹𝑥) < 𝑎 ↔ (𝐹𝑥) < 𝐴))
109rabbidv 3177 . . . 4 (𝑎 = 𝐴 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴})
1110eleq1d 2683 . . 3 (𝑎 = 𝐴 → ({𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷)))
1211rspcva 3293 . 2 ((𝐴 ∈ ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))
131, 8, 12syl2anc 692 1 (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  wnfc 2748  wral 2907  {crab 2911  wss 3555   cuni 4402   class class class wbr 4613  dom cdm 5074  wf 5843  cfv 5847  (class class class)co 6604  cr 9879   < clt 10018  t crest 16002  SAlgcsalg 39832  SMblFncsmblfn 40213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954  ax-pre-lttrn 9955
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-ioo 12121  df-ico 12123  df-smblfn 40214
This theorem is referenced by:  smfpimltmpt  40259  smfpimltxr  40260
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