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Theorem salpreimagelt 45022
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 2746 . . . . 5 𝑆 = 𝐴
32a1i 11 . . . 4 (𝜑 𝑆 = 𝐴)
43difeq1d 4086 . . 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 11212 . . . 4 (𝜑𝐶 ∈ ℝ*)
95, 6, 8preimagelt 45014 . . 3 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶})
104, 9eqtr2d 2778 . 2 (𝜑 → {𝑥𝐴𝐵 < 𝐶} = ( 𝑆 ∖ {𝑥𝐴𝐶𝐵}))
11 salpreimagelt.s . . 3 (𝜑𝑆 ∈ SAlg)
127ancli 550 . . . 4 (𝜑 → (𝜑𝐶 ∈ ℝ))
13 salpreimagelt.a . . . . . . 7 𝑎𝜑
14 nfcv 2908 . . . . . . . 8 𝑎𝐶
1514nfel1 2924 . . . . . . 7 𝑎 𝐶 ∈ ℝ
1613, 15nfan 1903 . . . . . 6 𝑎(𝜑𝐶 ∈ ℝ)
17 nfv 1918 . . . . . 6 𝑎{𝑥𝐴𝐶𝐵} ∈ 𝑆
1816, 17nfim 1900 . . . . 5 𝑎((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐶𝐵} ∈ 𝑆)
19 eleq1 2826 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ))
2019anbi2d 630 . . . . . 6 (𝑎 = 𝐶 → ((𝜑𝑎 ∈ ℝ) ↔ (𝜑𝐶 ∈ ℝ)))
21 breq1 5113 . . . . . . . 8 (𝑎 = 𝐶 → (𝑎𝐵𝐶𝐵))
2221rabbidv 3418 . . . . . . 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 3527 . . . 4 (𝐶 ∈ ℝ → ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐶𝐵} ∈ 𝑆))
277, 12, 26sylc 65 . . 3 (𝜑 → {𝑥𝐴𝐶𝐵} ∈ 𝑆)
2811, 27saldifcld 44662 . 2 (𝜑 → ( 𝑆 ∖ {𝑥𝐴𝐶𝐵}) ∈ 𝑆)
2910, 28eqeltrd 2838 1 (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wnf 1786  wcel 2107  {crab 3410  cdif 3912   cuni 4870   class class class wbr 5110  cr 11057  *cxr 11195   < clt 11196  cle 11197  SAlgcsalg 44623
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-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  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-br 5111  df-opab 5173  df-xp 5644  df-cnv 5646  df-xr 11200  df-le 11202  df-salg 44624
This theorem is referenced by:  salpreimalelt  45044  salpreimagtlt  45045  issmfgelem  45084
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