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Theorem issmff 45061
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 β„²π‘₯𝐹
issmff.s (πœ‘ β†’ 𝑆 ∈ SAlg)
issmff.d 𝐷 = dom 𝐹
Assertion
Ref Expression
issmff (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   𝐹,π‘Ž   𝑆,π‘Ž   π‘₯,π‘Ž
Allowed substitution hints:   πœ‘(π‘₯,π‘Ž)   𝐷(π‘₯)   𝑆(π‘₯)   𝐹(π‘₯)

Proof of Theorem issmff
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 issmff.s . . 3 (πœ‘ β†’ 𝑆 ∈ SAlg)
2 issmff.d . . 3 𝐷 = dom 𝐹
31, 2issmf 45055 . 2 (πœ‘ β†’ (𝐹 ∈ (SMblFnβ€˜π‘†) ↔ (𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))))
4 nfcv 2904 . . . . . . 7 Ⅎ𝑦𝐷
5 issmff.x . . . . . . . . 9 β„²π‘₯𝐹
65nfdm 5907 . . . . . . . 8 β„²π‘₯dom 𝐹
72, 6nfcxfr 2902 . . . . . . 7 β„²π‘₯𝐷
8 nfcv 2904 . . . . . . . . 9 β„²π‘₯𝑦
95, 8nffv 6853 . . . . . . . 8 β„²π‘₯(πΉβ€˜π‘¦)
10 nfcv 2904 . . . . . . . 8 β„²π‘₯ <
11 nfcv 2904 . . . . . . . 8 β„²π‘₯π‘Ž
129, 10, 11nfbr 5153 . . . . . . 7 β„²π‘₯(πΉβ€˜π‘¦) < π‘Ž
13 nfv 1918 . . . . . . 7 Ⅎ𝑦(πΉβ€˜π‘₯) < π‘Ž
14 fveq2 6843 . . . . . . . 8 (𝑦 = π‘₯ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯))
1514breq1d 5116 . . . . . . 7 (𝑦 = π‘₯ β†’ ((πΉβ€˜π‘¦) < π‘Ž ↔ (πΉβ€˜π‘₯) < π‘Ž))
164, 7, 12, 13, 15cbvrabw 3438 . . . . . 6 {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž} = {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž}
1716eleq1i 2825 . . . . 5 ({𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷) ↔ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))
1817ralbii 3093 . . . 4 (βˆ€π‘Ž ∈ ℝ {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷) ↔ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))
19183anbi3i 1160 . . 3 ((𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž} ∈ (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)))
2019a1i 11 . 2 (πœ‘ β†’ ((𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž} ∈ (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))))
213, 20bitrd 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 1088   = wceq 1542   ∈ wcel 2107  β„²wnfc 2884  βˆ€wral 3061  {crab 3406   βŠ† wss 3911  βˆͺ cuni 4866   class class class wbr 5106  dom cdm 5634  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  β„cr 11055   < clt 11194   β†Ύt crest 17307  SAlgcsalg 44635  SMblFncsmblfn 45022
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-pre-lttri 11130  ax-pre-lttrn 11131
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-er 8651  df-pm 8771  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-ioo 13274  df-ico 13276  df-smblfn 45023
This theorem is referenced by:  smfpreimaltf  45063  issmfdf  45064
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