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Theorem issmff 46391
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 46385 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷))))
4 nfcv 2892 . . . . . . 7 𝑦𝐷
5 issmff.x . . . . . . . . 9 𝑥𝐹
65nfdm 5949 . . . . . . . 8 𝑥dom 𝐹
72, 6nfcxfr 2890 . . . . . . 7 𝑥𝐷
8 nfcv 2892 . . . . . . . . 9 𝑥𝑦
95, 8nffv 6903 . . . . . . . 8 𝑥(𝐹𝑦)
10 nfcv 2892 . . . . . . . 8 𝑥 <
11 nfcv 2892 . . . . . . . 8 𝑥𝑎
129, 10, 11nfbr 5192 . . . . . . 7 𝑥(𝐹𝑦) < 𝑎
13 nfv 1910 . . . . . . 7 𝑦(𝐹𝑥) < 𝑎
14 fveq2 6893 . . . . . . . 8 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
1514breq1d 5155 . . . . . . 7 (𝑦 = 𝑥 → ((𝐹𝑦) < 𝑎 ↔ (𝐹𝑥) < 𝑎))
164, 7, 12, 13, 15cbvrabw 3456 . . . . . 6 {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} = {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎}
1716eleq1i 2817 . . . . 5 ({𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
1817ralbii 3083 . . . 4 (∀𝑎 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) < 𝑎} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
19183anbi3i 1156 . . 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 278 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 205  w3a 1084   = wceq 1534  wcel 2099  wnfc 2876  wral 3051  {crab 3419  wss 3946   cuni 4905   class class class wbr 5145  dom cdm 5674  wf 6542  cfv 6546  (class class class)co 7416  cr 11148   < clt 11289  t crest 17430  SAlgcsalg 45965  SMblFncsmblfn 46352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-pre-lttri 11223  ax-pre-lttrn 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-po 5586  df-so 5587  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7995  df-2nd 7996  df-er 8726  df-pm 8850  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-ioo 13376  df-ico 13378  df-smblfn 46353
This theorem is referenced by:  smfpreimaltf  46393  issmfdf  46394
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