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Theorem smfpimltmpt 40292
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 4717 . . 3 𝑥(𝑥𝐴𝐵)
2 smfpimltmpt.s . . 3 (𝜑𝑆 ∈ SAlg)
3 smfpimltmpt.f . . 3 (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
4 eqid 2621 . . 3 dom (𝑥𝐴𝐵) = dom (𝑥𝐴𝐵)
5 smfpimltmpt.r . . 3 (𝜑𝑅 ∈ ℝ)
61, 2, 3, 4, 5smfpreimaltf 40282 . 2 (𝜑 → {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} ∈ (𝑆t dom (𝑥𝐴𝐵)))
7 smfpimltmpt.x . . . . . 6 𝑥𝜑
8 eqid 2621 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
9 smfpimltmpt.b . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑉)
107, 8, 9dmmptdf 38926 . . . . 5 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
111nfdm 5337 . . . . . 6 𝑥dom (𝑥𝐴𝐵)
12 nfcv 2761 . . . . . 6 𝑥𝐴
1311, 12rabeqf 3182 . . . . 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 6260 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1716breq1d 4633 . . . . 5 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥) < 𝑅𝐵 < 𝑅))
187, 17rabbida 38796 . . . 4 (𝜑 → {𝑥𝐴 ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} = {𝑥𝐴𝐵 < 𝑅})
19 eqidd 2622 . . . 4 (𝜑 → {𝑥𝐴𝐵 < 𝑅} = {𝑥𝐴𝐵 < 𝑅})
2014, 18, 193eqtrrd 2660 . . 3 (𝜑 → {𝑥𝐴𝐵 < 𝑅} = {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅})
2110eqcomd 2627 . . . 4 (𝜑𝐴 = dom (𝑥𝐴𝐵))
2221oveq2d 6631 . . 3 (𝜑 → (𝑆t 𝐴) = (𝑆t dom (𝑥𝐴𝐵)))
2320, 22eleq12d 2692 . 2 (𝜑 → ({𝑥𝐴𝐵 < 𝑅} ∈ (𝑆t 𝐴) ↔ {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} ∈ (𝑆t dom (𝑥𝐴𝐵))))
246, 23mpbird 247 1 (𝜑 → {𝑥𝐴𝐵 < 𝑅} ∈ (𝑆t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wnf 1705  wcel 1987  {crab 2912   class class class wbr 4623  cmpt 4683  dom cdm 5084  cfv 5857  (class class class)co 6615  cr 9895   < clt 10034  t crest 16021  SAlgcsalg 39865  SMblFncsmblfn 40246
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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-pre-lttri 9970  ax-pre-lttrn 9971
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-po 5005  df-so 5006  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-er 7702  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-ioo 12137  df-ico 12139  df-smblfn 40247
This theorem is referenced by:  smfaddlem2  40309  smfrec  40333
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