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Theorem issmf 46116
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
issmf.s (πœ‘ β†’ 𝑆 ∈ SAlg)
issmf.d 𝐷 = dom 𝐹
Assertion
Ref Expression
issmf (πœ‘ β†’ (𝐹 ∈ (SMblFnβ€˜π‘†) ↔ (𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))))
Distinct variable groups:   𝐷,π‘Ž,π‘₯   𝐹,π‘Ž,π‘₯   𝑆,π‘Ž
Allowed substitution hints:   πœ‘(π‘₯,π‘Ž)   𝑆(π‘₯)

Proof of Theorem issmf
Dummy variables 𝑏 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issmf.s . . 3 (πœ‘ β†’ 𝑆 ∈ SAlg)
2 issmf.d . . 3 𝐷 = dom 𝐹
31, 2issmflem 46115 . 2 (πœ‘ β†’ (𝐹 ∈ (SMblFnβ€˜π‘†) ↔ (𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘ ∈ ℝ {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < 𝑏} ∈ (𝑆 β†Ύt 𝐷))))
4 breq2 5152 . . . . . . . 8 (𝑏 = π‘Ž β†’ ((πΉβ€˜π‘¦) < 𝑏 ↔ (πΉβ€˜π‘¦) < π‘Ž))
54rabbidv 3437 . . . . . . 7 (𝑏 = π‘Ž β†’ {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < 𝑏} = {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž})
65eleq1d 2814 . . . . . 6 (𝑏 = π‘Ž β†’ ({𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < 𝑏} ∈ (𝑆 β†Ύt 𝐷) ↔ {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷)))
7 fveq2 6897 . . . . . . . . . 10 (𝑦 = π‘₯ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯))
87breq1d 5158 . . . . . . . . 9 (𝑦 = π‘₯ β†’ ((πΉβ€˜π‘¦) < π‘Ž ↔ (πΉβ€˜π‘₯) < π‘Ž))
98cbvrabv 3439 . . . . . . . 8 {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž} = {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž}
109eleq1i 2820 . . . . . . 7 ({𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷) ↔ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))
1110a1i 11 . . . . . 6 (𝑏 = π‘Ž β†’ ({𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷) ↔ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷)))
126, 11bitrd 279 . . . . 5 (𝑏 = π‘Ž β†’ ({𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < 𝑏} ∈ (𝑆 β†Ύt 𝐷) ↔ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷)))
1312cbvralvw 3231 . . . 4 (βˆ€π‘ ∈ ℝ {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < 𝑏} ∈ (𝑆 β†Ύt 𝐷) ↔ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))
14133anbi3i 1157 . . 3 ((𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘ ∈ ℝ {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < 𝑏} ∈ (𝑆 β†Ύt 𝐷)) ↔ (𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷)))
1514a1i 11 . 2 (πœ‘ β†’ ((𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘ ∈ ℝ {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < 𝑏} ∈ (𝑆 β†Ύt 𝐷)) ↔ (𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))))
163, 15bitrd 279 1 (πœ‘ β†’ (𝐹 ∈ (SMblFnβ€˜π‘†) ↔ (𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  {crab 3429   βŠ† wss 3947  βˆͺ cuni 4908   class class class wbr 5148  dom cdm 5678  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420  β„cr 11137   < clt 11278   β†Ύt crest 17401  SAlgcsalg 45696  SMblFncsmblfn 46083
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 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11194  ax-resscn 11195  ax-pre-lttri 11212  ax-pre-lttrn 11213
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-er 8724  df-pm 8847  df-en 8964  df-dom 8965  df-sdom 8966  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-ioo 13360  df-ico 13362  df-smblfn 46084
This theorem is referenced by:  smfpreimalt  46119  smff  46120  smfdmss  46121  issmff  46122  issmfd  46123  issmflelem  46132  issmfgtlem  46143  issmfgelem  46157
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