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Theorem salpreimalegt 39396
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 2613 . . . . 5 𝑆 = 𝐴
32a1i 11 . . . 4 (𝜑 𝑆 = 𝐴)
43difeq1d 3683 . . 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 9940 . . . 4 (𝜑𝐶 ∈ ℝ*)
95, 6, 8preimalegt 39389 . . 3 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐵𝐶}) = {𝑥𝐴𝐶 < 𝐵})
104, 9eqtr2d 2639 . 2 (𝜑 → {𝑥𝐴𝐶 < 𝐵} = ( 𝑆 ∖ {𝑥𝐴𝐵𝐶}))
11 salpreimalegt.s . . 3 (𝜑𝑆 ∈ SAlg)
127ancli 571 . . . 4 (𝜑 → (𝜑𝐶 ∈ ℝ))
13 nfcv 2745 . . . . 5 𝑎𝐶
14 salpreimalegt.a . . . . . . 7 𝑎𝜑
15 nfv 1828 . . . . . . 7 𝑎 𝐶 ∈ ℝ
1614, 15nfan 1814 . . . . . 6 𝑎(𝜑𝐶 ∈ ℝ)
17 nfv 1828 . . . . . 6 𝑎{𝑥𝐴𝐵𝐶} ∈ 𝑆
1816, 17nfim 1811 . . . . 5 𝑎((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐵𝐶} ∈ 𝑆)
19 eleq1 2670 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ))
2019anbi2d 735 . . . . . 6 (𝑎 = 𝐶 → ((𝜑𝑎 ∈ ℝ) ↔ (𝜑𝐶 ∈ ℝ)))
21 breq2 4576 . . . . . . . 8 (𝑎 = 𝐶 → (𝐵𝑎𝐵𝐶))
2221rabbidv 3158 . . . . . . 7 (𝑎 = 𝐶 → {𝑥𝐴𝐵𝑎} = {𝑥𝐴𝐵𝐶})
2322eleq1d 2666 . . . . . 6 (𝑎 = 𝐶 → ({𝑥𝐴𝐵𝑎} ∈ 𝑆 ↔ {𝑥𝐴𝐵𝐶} ∈ 𝑆))
2420, 23imbi12d 332 . . . . 5 (𝑎 = 𝐶 → (((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆) ↔ ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐵𝐶} ∈ 𝑆)))
25 salpreimalegt.p . . . . 5 ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆)
2613, 18, 24, 25vtoclgf 3231 . . . 4 (𝐶 ∈ ℝ → ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐵𝐶} ∈ 𝑆))
277, 12, 26sylc 62 . . 3 (𝜑 → {𝑥𝐴𝐵𝐶} ∈ 𝑆)
2811, 27saldifcld 39040 . 2 (𝜑 → ( 𝑆 ∖ {𝑥𝐴𝐵𝐶}) ∈ 𝑆)
2910, 28eqeltrd 2682 1 (𝜑 → {𝑥𝐴𝐶 < 𝐵} ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wnf 1698  wcel 1975  {crab 2894  cdif 3531   cuni 4361   class class class wbr 4572  cr 9786  *cxr 9924   < clt 9925  cle 9926  SAlgcsalg 39003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-xp 5029  df-cnv 5031  df-xr 9929  df-le 9931  df-salg 39004
This theorem is referenced by:  salpreimalelt  39414  issmfgt  39442
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