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Theorem smfpimltmpt 41475
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 4881 . . 3 𝑥(𝑥𝐴𝐵)
2 smfpimltmpt.s . . 3 (𝜑𝑆 ∈ SAlg)
3 smfpimltmpt.f . . 3 (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
4 eqid 2771 . . 3 dom (𝑥𝐴𝐵) = dom (𝑥𝐴𝐵)
5 smfpimltmpt.r . . 3 (𝜑𝑅 ∈ ℝ)
61, 2, 3, 4, 5smfpreimaltf 41465 . 2 (𝜑 → {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} ∈ (𝑆t dom (𝑥𝐴𝐵)))
7 smfpimltmpt.x . . . . . 6 𝑥𝜑
8 eqid 2771 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
9 smfpimltmpt.b . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑉)
107, 8, 9dmmptdf 39935 . . . . 5 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
111nfdm 5505 . . . . . 6 𝑥dom (𝑥𝐴𝐵)
12 nfcv 2913 . . . . . 6 𝑥𝐴
1311, 12rabeqf 3340 . . . . 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 6435 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1716breq1d 4796 . . . . 5 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥) < 𝑅𝐵 < 𝑅))
187, 17rabbida 39795 . . . 4 (𝜑 → {𝑥𝐴 ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅} = {𝑥𝐴𝐵 < 𝑅})
19 eqidd 2772 . . . 4 (𝜑 → {𝑥𝐴𝐵 < 𝑅} = {𝑥𝐴𝐵 < 𝑅})
2014, 18, 193eqtrrd 2810 . . 3 (𝜑 → {𝑥𝐴𝐵 < 𝑅} = {𝑥 ∈ dom (𝑥𝐴𝐵) ∣ ((𝑥𝐴𝐵)‘𝑥) < 𝑅})
2110eqcomd 2777 . . . 4 (𝜑𝐴 = dom (𝑥𝐴𝐵))
2221oveq2d 6809 . . 3 (𝜑 → (𝑆t 𝐴) = (𝑆t dom (𝑥𝐴𝐵)))
2320, 22eleq12d 2844 . 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 382   = wceq 1631  wnf 1856  wcel 2145  {crab 3065   class class class wbr 4786  cmpt 4863  dom cdm 5249  cfv 6031  (class class class)co 6793  cr 10137   < clt 10276  t crest 16289  SAlgcsalg 41045  SMblFncsmblfn 41429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-pre-lttri 10212  ax-pre-lttrn 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-er 7896  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-ioo 12384  df-ico 12386  df-smblfn 41430
This theorem is referenced by:  smfaddlem2  41492  smfrec  41516
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