Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  smfpreimaltf Structured version   Visualization version   GIF version

Theorem smfpreimaltf 44272
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 44270 . . . 4 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
72, 6mpbid 231 . . 3 (𝜑 → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)))
87simp3d 1143 . 2 (𝜑 → ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
9 breq2 5078 . . . . 5 (𝑎 = 𝐴 → ((𝐹𝑥) < 𝑎 ↔ (𝐹𝑥) < 𝐴))
109rabbidv 3414 . . . 4 (𝑎 = 𝐴 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴})
1110eleq1d 2823 . . 3 (𝑎 = 𝐴 → ({𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷)))
1211rspcva 3559 . 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 1086   = wceq 1539  wcel 2106  wnfc 2887  wral 3064  {crab 3068  wss 3887   cuni 4839   class class class wbr 5074  dom cdm 5589  wf 6429  cfv 6433  (class class class)co 7275  cr 10870   < clt 11009  t crest 17131  SAlgcsalg 43849  SMblFncsmblfn 44233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-pre-lttri 10945  ax-pre-lttrn 10946
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-er 8498  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-ioo 13083  df-ico 13085  df-smblfn 44234
This theorem is referenced by:  smfpimltmpt  44282  smfpimltxr  44283
  Copyright terms: Public domain W3C validator