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Theorem salpreimalegt 44920
Description: If all the preimages of right-closed, unbounded below intervals, belong to a sigma-algebra, then all the preimages of left-open, unbounded above intervals, belong to the sigma-algebra. (ii) implies (iii) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
salpreimalegt.x 𝑥𝜑
salpreimalegt.a 𝑎𝜑
salpreimalegt.s (𝜑𝑆 ∈ SAlg)
salpreimalegt.u 𝐴 = 𝑆
salpreimalegt.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
salpreimalegt.p ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆)
salpreimalegt.c (𝜑𝐶 ∈ ℝ)
Assertion
Ref Expression
salpreimalegt (𝜑 → {𝑥𝐴𝐶 < 𝐵} ∈ 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 salpreimalegt
StepHypRef Expression
1 salpreimalegt.u . . . . . 6 𝐴 = 𝑆
21eqcomi 2745 . . . . 5 𝑆 = 𝐴
32a1i 11 . . . 4 (𝜑 𝑆 = 𝐴)
43difeq1d 4080 . . 3 (𝜑 → ( 𝑆 ∖ {𝑥𝐴𝐵𝐶}) = (𝐴 ∖ {𝑥𝐴𝐵𝐶}))
5 salpreimalegt.x . . . 4 𝑥𝜑
6 salpreimalegt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
7 salpreimalegt.c . . . . 5 (𝜑𝐶 ∈ ℝ)
87rexrd 11202 . . . 4 (𝜑𝐶 ∈ ℝ*)
95, 6, 8preimalegt 44911 . . 3 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐵𝐶}) = {𝑥𝐴𝐶 < 𝐵})
104, 9eqtr2d 2777 . 2 (𝜑 → {𝑥𝐴𝐶 < 𝐵} = ( 𝑆 ∖ {𝑥𝐴𝐵𝐶}))
11 salpreimalegt.s . . 3 (𝜑𝑆 ∈ SAlg)
127ancli 549 . . . 4 (𝜑 → (𝜑𝐶 ∈ ℝ))
13 salpreimalegt.a . . . . . . 7 𝑎𝜑
14 nfv 1917 . . . . . . 7 𝑎 𝐶 ∈ ℝ
1513, 14nfan 1902 . . . . . 6 𝑎(𝜑𝐶 ∈ ℝ)
16 nfv 1917 . . . . . 6 𝑎{𝑥𝐴𝐵𝐶} ∈ 𝑆
1715, 16nfim 1899 . . . . 5 𝑎((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐵𝐶} ∈ 𝑆)
18 eleq1 2825 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ))
1918anbi2d 629 . . . . . 6 (𝑎 = 𝐶 → ((𝜑𝑎 ∈ ℝ) ↔ (𝜑𝐶 ∈ ℝ)))
20 breq2 5108 . . . . . . . 8 (𝑎 = 𝐶 → (𝐵𝑎𝐵𝐶))
2120rabbidv 3414 . . . . . . 7 (𝑎 = 𝐶 → {𝑥𝐴𝐵𝑎} = {𝑥𝐴𝐵𝐶})
2221eleq1d 2822 . . . . . 6 (𝑎 = 𝐶 → ({𝑥𝐴𝐵𝑎} ∈ 𝑆 ↔ {𝑥𝐴𝐵𝐶} ∈ 𝑆))
2319, 22imbi12d 344 . . . . 5 (𝑎 = 𝐶 → (((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆) ↔ ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐵𝐶} ∈ 𝑆)))
24 salpreimalegt.p . . . . 5 ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆)
2517, 23, 24vtoclg1f 3523 . . . 4 (𝐶 ∈ ℝ → ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐵𝐶} ∈ 𝑆))
267, 12, 25sylc 65 . . 3 (𝜑 → {𝑥𝐴𝐵𝐶} ∈ 𝑆)
2711, 26saldifcld 44558 . 2 (𝜑 → ( 𝑆 ∖ {𝑥𝐴𝐵𝐶}) ∈ 𝑆)
2810, 27eqeltrd 2838 1 (𝜑 → {𝑥𝐴𝐶 < 𝐵} ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wnf 1785  wcel 2106  {crab 3406  cdif 3906   cuni 4864   class class class wbr 5104  cr 11047  *cxr 11185   < clt 11186  cle 11187  SAlgcsalg 44519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-xp 5638  df-cnv 5640  df-xr 11190  df-le 11192  df-salg 44520
This theorem is referenced by:  salpreimalelt  44940  issmfgt  44967
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