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Theorem smfpimltmpt 46761
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
smfpimltmpt.x 𝑥𝜑
smfpimltmpt.s (𝜑𝑆 ∈ SAlg)
smfpimltmpt.b ((𝜑𝑥𝐴) → 𝐵𝑉)
smfpimltmpt.f (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
smfpimltmpt.r (𝜑𝑅 ∈ ℝ)
Assertion
Ref Expression
smfpimltmpt (𝜑 → {𝑥𝐴𝐵 < 𝑅} ∈ (𝑆t 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑆(𝑥)   𝑉(𝑥)

Proof of Theorem smfpimltmpt
StepHypRef Expression
1 nfmpt1 5250 . . 3 𝑥(𝑥𝐴𝐵)
2 smfpimltmpt.s . . 3 (𝜑𝑆 ∈ SAlg)
3 smfpimltmpt.f . . 3 (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
4 eqid 2737 . . 3 dom (𝑥𝐴𝐵) = dom (𝑥𝐴𝐵)
5 smfpimltmpt.r . . 3 (𝜑𝑅 ∈ ℝ)
61, 2, 3, 4, 5smfpreimaltf 46751 . 2 (𝜑 → {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} ∈ (𝑆t dom (𝑥𝐴𝐵)))
7 smfpimltmpt.x . . . . . 6 𝑥𝜑
8 eqid 2737 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
9 smfpimltmpt.b . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑉)
107, 8, 9dmmptdf 45229 . . . . 5 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
111nfdm 5962 . . . . . 6 𝑥dom (𝑥𝐴𝐵)
12 nfcv 2905 . . . . . 6 𝑥𝐴
1311, 12rabeqf 3472 . . . . 5 (dom (𝑥𝐴𝐵) = 𝐴 → {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} = {𝑥𝐴 ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅})
1410, 13syl 17 . . . 4 (𝜑 → {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} = {𝑥𝐴 ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅})
158a1i 11 . . . . . . 7 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
1615, 9fvmpt2d 7029 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1716breq1d 5153 . . . . 5 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥) < 𝑅𝐵 < 𝑅))
187, 17rabbida 3463 . . . 4 (𝜑 → {𝑥𝐴 ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} = {𝑥𝐴𝐵 < 𝑅})
19 eqidd 2738 . . . 4 (𝜑 → {𝑥𝐴𝐵 < 𝑅} = {𝑥𝐴𝐵 < 𝑅})
2014, 18, 193eqtrrd 2782 . . 3 (𝜑 → {𝑥𝐴𝐵 < 𝑅} = {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅})
2110eqcomd 2743 . . . 4 (𝜑𝐴 = dom (𝑥𝐴𝐵))
2221oveq2d 7447 . . 3 (𝜑 → (𝑆t 𝐴) = (𝑆t dom (𝑥𝐴𝐵)))
2320, 22eleq12d 2835 . 2 (𝜑 → ({𝑥𝐴𝐵 < 𝑅} ∈ (𝑆t 𝐴) ↔ {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} ∈ (𝑆t dom (𝑥𝐴𝐵))))
246, 23mpbird 257 1 (𝜑 → {𝑥𝐴𝐵 < 𝑅} ∈ (𝑆t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wnf 1783  wcel 2108  {crab 3436   class class class wbr 5143  cmpt 5225  dom cdm 5685  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:  smfaddlem2  46779  smfrec  46804
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