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Theorem smfpreimaltf 43833
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
smfpreimaltf.x 𝑥𝐹
smfpreimaltf.s (𝜑𝑆 ∈ SAlg)
smfpreimaltf.f (𝜑𝐹 ∈ (SMblFn‘𝑆))
smfpreimaltf.d 𝐷 = dom 𝐹
smfpreimaltf.a (𝜑𝐴 ∈ ℝ)
Assertion
Ref Expression
smfpreimaltf (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐷(𝑥)   𝑆(𝑥)   𝐹(𝑥)

Proof of Theorem smfpreimaltf
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 smfpreimaltf.a . 2 (𝜑𝐴 ∈ ℝ)
2 smfpreimaltf.f . . . 4 (𝜑𝐹 ∈ (SMblFn‘𝑆))
3 smfpreimaltf.x . . . . 5 𝑥𝐹
4 smfpreimaltf.s . . . . 5 (𝜑𝑆 ∈ SAlg)
5 smfpreimaltf.d . . . . 5 𝐷 = dom 𝐹
63, 4, 5issmff 43831 . . . 4 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
72, 6mpbid 235 . . 3 (𝜑 → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)))
87simp3d 1145 . 2 (𝜑 → ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
9 breq2 5034 . . . . 5 (𝑎 = 𝐴 → ((𝐹𝑥) < 𝑎 ↔ (𝐹𝑥) < 𝐴))
109rabbidv 3381 . . . 4 (𝑎 = 𝐴 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴})
1110eleq1d 2817 . . 3 (𝑎 = 𝐴 → ({𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷)))
1211rspcva 3524 . 2 ((𝐴 ∈ ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))
131, 8, 12syl2anc 587 1 (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2114  wnfc 2879  wral 3053  {crab 3057  wss 3843   cuni 4796   class class class wbr 5030  dom cdm 5525  wf 6335  cfv 6339  (class class class)co 7172  cr 10616   < clt 10755  t crest 16799  SAlgcsalg 43413  SMblFncsmblfn 43797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481  ax-cnex 10673  ax-resscn 10674  ax-pre-lttri 10691  ax-pre-lttrn 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-po 5442  df-so 5443  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7175  df-oprab 7176  df-mpo 7177  df-1st 7716  df-2nd 7717  df-er 8322  df-pm 8442  df-en 8558  df-dom 8559  df-sdom 8560  df-pnf 10757  df-mnf 10758  df-xr 10759  df-ltxr 10760  df-le 10761  df-ioo 12827  df-ico 12829  df-smblfn 43798
This theorem is referenced by:  smfpimltmpt  43843  smfpimltxr  43844
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