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Theorem smfpimltmpt 46702
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 5256 . . 3 𝑥(𝑥𝐴𝐵)
2 smfpimltmpt.s . . 3 (𝜑𝑆 ∈ SAlg)
3 smfpimltmpt.f . . 3 (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
4 eqid 2735 . . 3 dom (𝑥𝐴𝐵) = dom (𝑥𝐴𝐵)
5 smfpimltmpt.r . . 3 (𝜑𝑅 ∈ ℝ)
61, 2, 3, 4, 5smfpreimaltf 46692 . 2 (𝜑 → {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} ∈ (𝑆t dom (𝑥𝐴𝐵)))
7 smfpimltmpt.x . . . . . 6 𝑥𝜑
8 eqid 2735 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
9 smfpimltmpt.b . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑉)
107, 8, 9dmmptdf 45167 . . . . 5 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
111nfdm 5965 . . . . . 6 𝑥dom (𝑥𝐴𝐵)
12 nfcv 2903 . . . . . 6 𝑥𝐴
1311, 12rabeqf 3470 . . . . 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 5158 . . . . 5 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥) < 𝑅𝐵 < 𝑅))
187, 17rabbida 3461 . . . 4 (𝜑 → {𝑥𝐴 ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} = {𝑥𝐴𝐵 < 𝑅})
19 eqidd 2736 . . . 4 (𝜑 → {𝑥𝐴𝐵 < 𝑅} = {𝑥𝐴𝐵 < 𝑅})
2014, 18, 193eqtrrd 2780 . . 3 (𝜑 → {𝑥𝐴𝐵 < 𝑅} = {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅})
2110eqcomd 2741 . . . 4 (𝜑𝐴 = dom (𝑥𝐴𝐵))
2221oveq2d 7447 . . 3 (𝜑 → (𝑆t 𝐴) = (𝑆t dom (𝑥𝐴𝐵)))
2320, 22eleq12d 2833 . 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 1537  wnf 1780  wcel 2106  {crab 3433   class class class wbr 5148  cmpt 5231  dom cdm 5689  cfv 6563  (class class class)co 7431  cr 11152   < clt 11293  t crest 17467  SAlgcsalg 46264  SMblFncsmblfn 46651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-er 8744  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-ioo 13388  df-ico 13390  df-smblfn 46652
This theorem is referenced by:  smfaddlem2  46720  smfrec  46745
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