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Theorem issmff 43735
 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 43729 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷))))
4 nfcv 2920 . . . . . . 7 𝑦𝐷
5 issmff.x . . . . . . . . 9 𝑥𝐹
65nfdm 5793 . . . . . . . 8 𝑥dom 𝐹
72, 6nfcxfr 2918 . . . . . . 7 𝑥𝐷
8 nfcv 2920 . . . . . . . . 9 𝑥𝑦
95, 8nffv 6669 . . . . . . . 8 𝑥(𝐹𝑦)
10 nfcv 2920 . . . . . . . 8 𝑥 <
11 nfcv 2920 . . . . . . . 8 𝑥𝑎
129, 10, 11nfbr 5080 . . . . . . 7 𝑥(𝐹𝑦) < 𝑎
13 nfv 1916 . . . . . . 7 𝑦(𝐹𝑥) < 𝑎
14 fveq2 6659 . . . . . . . 8 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
1514breq1d 5043 . . . . . . 7 (𝑦 = 𝑥 → ((𝐹𝑦) < 𝑎 ↔ (𝐹𝑥) < 𝑎))
164, 7, 12, 13, 15cbvrabw 3403 . . . . . 6 {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎}
1716eleq1i 2843 . . . . 5 ({𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
1817ralbii 3098 . . . 4 (∀𝑎 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
19183anbi3i 1157 . . 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 282 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 209   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  Ⅎwnfc 2900  ∀wral 3071  {crab 3075   ⊆ wss 3859  ∪ cuni 4799   class class class wbr 5033  dom cdm 5525  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151  ℝcr 10575   < clt 10714   ↾t crest 16753  SAlgcsalg 43317  SMblFncsmblfn 43701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10632  ax-resscn 10633  ax-pre-lttri 10650  ax-pre-lttrn 10651 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-po 5444  df-so 5445  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 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7694  df-2nd 7695  df-er 8300  df-pm 8420  df-en 8529  df-dom 8530  df-sdom 8531  df-pnf 10716  df-mnf 10717  df-xr 10718  df-ltxr 10719  df-le 10720  df-ioo 12784  df-ico 12786  df-smblfn 43702 This theorem is referenced by:  smfpreimaltf  43737  issmfdf  43738  smfpimltxr  43748
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