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Theorem issmf 45043
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 45042 . 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 5114 . . . . . . . 8 (𝑏 = π‘Ž β†’ ((πΉβ€˜π‘¦) < 𝑏 ↔ (πΉβ€˜π‘¦) < π‘Ž))
54rabbidv 3418 . . . . . . 7 (𝑏 = π‘Ž β†’ {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < 𝑏} = {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž})
65eleq1d 2823 . . . . . 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 6847 . . . . . . . . . 10 (𝑦 = π‘₯ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯))
87breq1d 5120 . . . . . . . . 9 (𝑦 = π‘₯ β†’ ((πΉβ€˜π‘¦) < π‘Ž ↔ (πΉβ€˜π‘₯) < π‘Ž))
98cbvrabv 3420 . . . . . . . 8 {𝑦 ∈ 𝐷 ∣ (πΉβ€˜π‘¦) < π‘Ž} = {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž}
109eleq1i 2829 . . . . . . 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 3228 . . . 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 1160 . . 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 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410   βŠ† wss 3915  βˆͺ cuni 4870   class class class wbr 5110  dom cdm 5638  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  β„cr 11057   < clt 11196   β†Ύt crest 17309  SAlgcsalg 44623  SMblFncsmblfn 45010
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-pre-lttri 11132  ax-pre-lttrn 11133
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-po 5550  df-so 5551  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-er 8655  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-ioo 13275  df-ico 13277  df-smblfn 45011
This theorem is referenced by:  smfpreimalt  45046  smff  45047  smfdmss  45048  issmff  45049  issmfd  45050  issmflelem  45059  issmfgtlem  45070  issmfgelem  45084
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