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Theorem salpreimagelt 44201
Description: If all the preimages of left-closed, 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 2747 . . . . 5 𝑆 = 𝐴
32a1i 11 . . . 4 (𝜑 𝑆 = 𝐴)
43difeq1d 4056 . . 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 11013 . . . 4 (𝜑𝐶 ∈ ℝ*)
95, 6, 8preimagelt 44195 . . 3 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶})
104, 9eqtr2d 2779 . 2 (𝜑 → {𝑥𝐴𝐵 < 𝐶} = ( 𝑆 ∖ {𝑥𝐴𝐶𝐵}))
11 salpreimagelt.s . . 3 (𝜑𝑆 ∈ SAlg)
127ancli 549 . . . 4 (𝜑 → (𝜑𝐶 ∈ ℝ))
13 salpreimagelt.a . . . . . . 7 𝑎𝜑
14 nfcv 2907 . . . . . . . 8 𝑎𝐶
1514nfel1 2923 . . . . . . 7 𝑎 𝐶 ∈ ℝ
1613, 15nfan 1902 . . . . . 6 𝑎(𝜑𝐶 ∈ ℝ)
17 nfv 1917 . . . . . 6 𝑎{𝑥𝐴𝐶𝐵} ∈ 𝑆
1816, 17nfim 1899 . . . . 5 𝑎((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐶𝐵} ∈ 𝑆)
19 eleq1 2826 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ))
2019anbi2d 629 . . . . . 6 (𝑎 = 𝐶 → ((𝜑𝑎 ∈ ℝ) ↔ (𝜑𝐶 ∈ ℝ)))
21 breq1 5077 . . . . . . . 8 (𝑎 = 𝐶 → (𝑎𝐵𝐶𝐵))
2221rabbidv 3412 . . . . . . 7 (𝑎 = 𝐶 → {𝑥𝐴𝑎𝐵} = {𝑥𝐴𝐶𝐵})
2322eleq1d 2823 . . . . . 6 (𝑎 = 𝐶 → ({𝑥𝐴𝑎𝐵} ∈ 𝑆 ↔ {𝑥𝐴𝐶𝐵} ∈ 𝑆))
2420, 23imbi12d 345 . . . . 5 (𝑎 = 𝐶 → (((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎𝐵} ∈ 𝑆) ↔ ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐶𝐵} ∈ 𝑆)))
25 salpreimagelt.p . . . . 5 ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎𝐵} ∈ 𝑆)
2618, 24, 25vtoclg1f 3502 . . . 4 (𝐶 ∈ ℝ → ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐶𝐵} ∈ 𝑆))
277, 12, 26sylc 65 . . 3 (𝜑 → {𝑥𝐴𝐶𝐵} ∈ 𝑆)
2811, 27saldifcld 43845 . 2 (𝜑 → ( 𝑆 ∖ {𝑥𝐴𝐶𝐵}) ∈ 𝑆)
2910, 28eqeltrd 2839 1 (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wnf 1786  wcel 2106  {crab 3068  cdif 3884   cuni 4840   class class class wbr 5074  cr 10858  *cxr 10996   < clt 10997  cle 10998  SAlgcsalg 43808
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-opab 5137  df-xp 5591  df-cnv 5593  df-xr 11001  df-le 11003  df-salg 43809
This theorem is referenced by:  salpreimalelt  44221  salpreimagtlt  44222  issmfgelem  44260
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