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Theorem issmf 45989
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 45988 . 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 5143 . . . . . . . 8 (𝑏 = π‘Ž β†’ ((πΉβ€˜π‘¦) < 𝑏 ↔ (πΉβ€˜π‘¦) < π‘Ž))
54rabbidv 3432 . . . . . . 7 (𝑏 = π‘Ž β†’ {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < 𝑏} = {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž})
65eleq1d 2810 . . . . . 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 6882 . . . . . . . . . 10 (𝑦 = π‘₯ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯))
87breq1d 5149 . . . . . . . . 9 (𝑦 = π‘₯ β†’ ((πΉβ€˜π‘¦) < π‘Ž ↔ (πΉβ€˜π‘₯) < π‘Ž))
98cbvrabv 3434 . . . . . . . 8 {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž} = {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž}
109eleq1i 2816 . . . . . . 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 3226 . . . 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 1156 . . 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 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  {crab 3424   βŠ† wss 3941  βˆͺ cuni 4900   class class class wbr 5139  dom cdm 5667  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  β„cr 11106   < clt 11247   β†Ύt crest 17371  SAlgcsalg 45569  SMblFncsmblfn 45956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-pre-lttri 11181  ax-pre-lttrn 11182
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-po 5579  df-so 5580  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-er 8700  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-ioo 13329  df-ico 13331  df-smblfn 45957
This theorem is referenced by:  smfpreimalt  45992  smff  45993  smfdmss  45994  issmff  45995  issmfd  45996  issmflelem  46005  issmfgtlem  46016  issmfgelem  46030
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