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Theorem salpreimagelt 41239
Description: If all the preimages of left-close, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iv) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
salpreimagelt.x 𝑥𝜑
salpreimagelt.a 𝑎𝜑
salpreimagelt.s (𝜑𝑆 ∈ SAlg)
salpreimagelt.u 𝐴 = 𝑆
salpreimagelt.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
salpreimagelt.p ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎𝐵} ∈ 𝑆)
salpreimagelt.c (𝜑𝐶 ∈ ℝ)
Assertion
Ref Expression
salpreimagelt (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)
Distinct variable groups:   𝐴,𝑎,𝑥   𝐵,𝑎   𝐶,𝑎,𝑥   𝑆,𝑎
Allowed substitution hints:   𝜑(𝑥,𝑎)   𝐵(𝑥)   𝑆(𝑥)

Proof of Theorem salpreimagelt
StepHypRef Expression
1 salpreimagelt.u . . . . . 6 𝐴 = 𝑆
21eqcomi 2660 . . . . 5 𝑆 = 𝐴
32a1i 11 . . . 4 (𝜑 𝑆 = 𝐴)
43difeq1d 3760 . . 3 (𝜑 → ( 𝑆 ∖ {𝑥𝐴𝐶𝐵}) = (𝐴 ∖ {𝑥𝐴𝐶𝐵}))
5 salpreimagelt.x . . . 4 𝑥𝜑
6 salpreimagelt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
7 salpreimagelt.c . . . . 5 (𝜑𝐶 ∈ ℝ)
87rexrd 10127 . . . 4 (𝜑𝐶 ∈ ℝ*)
95, 6, 8preimagelt 41233 . . 3 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶})
104, 9eqtr2d 2686 . 2 (𝜑 → {𝑥𝐴𝐵 < 𝐶} = ( 𝑆 ∖ {𝑥𝐴𝐶𝐵}))
11 salpreimagelt.s . . 3 (𝜑𝑆 ∈ SAlg)
12 id 22 . . . . 5 (𝜑𝜑)
1312, 7jca 553 . . . 4 (𝜑 → (𝜑𝐶 ∈ ℝ))
14 nfcv 2793 . . . . 5 𝑎𝐶
15 salpreimagelt.a . . . . . . 7 𝑎𝜑
1614nfel1 2808 . . . . . . 7 𝑎 𝐶 ∈ ℝ
1715, 16nfan 1868 . . . . . 6 𝑎(𝜑𝐶 ∈ ℝ)
18 nfv 1883 . . . . . 6 𝑎{𝑥𝐴𝐶𝐵} ∈ 𝑆
1917, 18nfim 1865 . . . . 5 𝑎((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐶𝐵} ∈ 𝑆)
20 eleq1 2718 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ))
2120anbi2d 740 . . . . . 6 (𝑎 = 𝐶 → ((𝜑𝑎 ∈ ℝ) ↔ (𝜑𝐶 ∈ ℝ)))
22 breq1 4688 . . . . . . . 8 (𝑎 = 𝐶 → (𝑎𝐵𝐶𝐵))
2322rabbidv 3220 . . . . . . 7 (𝑎 = 𝐶 → {𝑥𝐴𝑎𝐵} = {𝑥𝐴𝐶𝐵})
2423eleq1d 2715 . . . . . 6 (𝑎 = 𝐶 → ({𝑥𝐴𝑎𝐵} ∈ 𝑆 ↔ {𝑥𝐴𝐶𝐵} ∈ 𝑆))
2521, 24imbi12d 333 . . . . 5 (𝑎 = 𝐶 → (((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎𝐵} ∈ 𝑆) ↔ ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐶𝐵} ∈ 𝑆)))
26 salpreimagelt.p . . . . 5 ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎𝐵} ∈ 𝑆)
2714, 19, 25, 26vtoclgf 3295 . . . 4 (𝐶 ∈ ℝ → ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐶𝐵} ∈ 𝑆))
287, 13, 27sylc 65 . . 3 (𝜑 → {𝑥𝐴𝐶𝐵} ∈ 𝑆)
2911, 28saldifcld 40883 . 2 (𝜑 → ( 𝑆 ∖ {𝑥𝐴𝐶𝐵}) ∈ 𝑆)
3010, 29eqeltrd 2730 1 (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wnf 1748  wcel 2030  {crab 2945  cdif 3604   cuni 4468   class class class wbr 4685  cr 9973  *cxr 10111   < clt 10112  cle 10113  SAlgcsalg 40846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-xr 10116  df-le 10118  df-salg 40847
This theorem is referenced by:  salpreimalelt  41259  salpreimagtlt  41260  issmfgelem  41298
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