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Theorem smfpimltmpt 43746
 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 5130 . . 3 𝑥(𝑥𝐴𝐵)
2 smfpimltmpt.s . . 3 (𝜑𝑆 ∈ SAlg)
3 smfpimltmpt.f . . 3 (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
4 eqid 2758 . . 3 dom (𝑥𝐴𝐵) = dom (𝑥𝐴𝐵)
5 smfpimltmpt.r . . 3 (𝜑𝑅 ∈ ℝ)
61, 2, 3, 4, 5smfpreimaltf 43736 . 2 (𝜑 → {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} ∈ (𝑆t dom (𝑥𝐴𝐵)))
7 smfpimltmpt.x . . . . . 6 𝑥𝜑
8 eqid 2758 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
9 smfpimltmpt.b . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑉)
107, 8, 9dmmptdf 42222 . . . . 5 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
111nfdm 5792 . . . . . 6 𝑥dom (𝑥𝐴𝐵)
12 nfcv 2919 . . . . . 6 𝑥𝐴
1311, 12rabeqf 3393 . . . . 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 6772 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1716breq1d 5042 . . . . 5 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥) < 𝑅𝐵 < 𝑅))
187, 17rabbida 3386 . . . 4 (𝜑 → {𝑥𝐴 ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} = {𝑥𝐴𝐵 < 𝑅})
19 eqidd 2759 . . . 4 (𝜑 → {𝑥𝐴𝐵 < 𝑅} = {𝑥𝐴𝐵 < 𝑅})
2014, 18, 193eqtrrd 2798 . . 3 (𝜑 → {𝑥𝐴𝐵 < 𝑅} = {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅})
2110eqcomd 2764 . . . 4 (𝜑𝐴 = dom (𝑥𝐴𝐵))
2221oveq2d 7166 . . 3 (𝜑 → (𝑆t 𝐴) = (𝑆t dom (𝑥𝐴𝐵)))
2320, 22eleq12d 2846 . 2 (𝜑 → ({𝑥𝐴𝐵 < 𝑅} ∈ (𝑆t 𝐴) ↔ {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} ∈ (𝑆t dom (𝑥𝐴𝐵))))
246, 23mpbird 260 1 (𝜑 → {𝑥𝐴𝐵 < 𝑅} ∈ (𝑆t 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  {crab 3074   class class class wbr 5032   ↦ cmpt 5112  dom cdm 5524  ‘cfv 6335  (class class class)co 7150  ℝcr 10574   < clt 10713   ↾t crest 16752  SAlgcsalg 43316  SMblFncsmblfn 43700 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-pre-lttri 10649  ax-pre-lttrn 10650 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-po 5443  df-so 5444  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-er 8299  df-pm 8419  df-en 8528  df-dom 8529  df-sdom 8530  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-ioo 12783  df-ico 12785  df-smblfn 43701 This theorem is referenced by:  smfaddlem2  43763  smfrec  43787
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