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Theorem issmff 41607
Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
issmff.x 𝑥𝐹
issmff.s (𝜑𝑆 ∈ SAlg)
issmff.d 𝐷 = dom 𝐹
Assertion
Ref Expression
issmff (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
Distinct variable groups:   𝐷,𝑎   𝐹,𝑎   𝑆,𝑎   𝑥,𝑎
Allowed substitution hints:   𝜑(𝑥,𝑎)   𝐷(𝑥)   𝑆(𝑥)   𝐹(𝑥)

Proof of Theorem issmff
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 issmff.s . . 3 (𝜑𝑆 ∈ SAlg)
2 issmff.d . . 3 𝐷 = dom 𝐹
31, 2issmf 41601 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷))))
4 nfcv 2907 . . . . . . 7 𝑦𝐷
5 issmff.x . . . . . . . . 9 𝑥𝐹
65nfdm 5538 . . . . . . . 8 𝑥dom 𝐹
72, 6nfcxfr 2905 . . . . . . 7 𝑥𝐷
8 nfcv 2907 . . . . . . . . 9 𝑥𝑦
95, 8nffv 6389 . . . . . . . 8 𝑥(𝐹𝑦)
10 nfcv 2907 . . . . . . . 8 𝑥 <
11 nfcv 2907 . . . . . . . 8 𝑥𝑎
129, 10, 11nfbr 4858 . . . . . . 7 𝑥(𝐹𝑦) < 𝑎
13 nfv 2009 . . . . . . 7 𝑦(𝐹𝑥) < 𝑎
14 fveq2 6379 . . . . . . . 8 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
1514breq1d 4821 . . . . . . 7 (𝑦 = 𝑥 → ((𝐹𝑦) < 𝑎 ↔ (𝐹𝑥) < 𝑎))
164, 7, 12, 13, 15cbvrab 3347 . . . . . 6 {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎}
1716eleq1i 2835 . . . . 5 ({𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
1817ralbii 3127 . . . 4 (∀𝑎 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
19183anbi3i 1198 . . 3 ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷)) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)))
2019a1i 11 . 2 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷)) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
213, 20bitrd 270 1 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  w3a 1107   = wceq 1652  wcel 2155  wnfc 2894  wral 3055  {crab 3059  wss 3734   cuni 4596   class class class wbr 4811  dom cdm 5279  wf 6066  cfv 6070  (class class class)co 6846  cr 10192   < clt 10332  t crest 16361  SAlgcsalg 41189  SMblFncsmblfn 41573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-pre-lttri 10267  ax-pre-lttrn 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-po 5200  df-so 5201  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-1st 7370  df-2nd 7371  df-er 7951  df-pm 8067  df-en 8165  df-dom 8166  df-sdom 8167  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-ioo 12386  df-ico 12388  df-smblfn 41574
This theorem is referenced by:  smfpreimaltf  41609  issmfdf  41610  smfpimltxr  41620
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