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Theorem salpreimalegt 42865
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 2827 . . . . 5 𝑆 = 𝐴
32a1i 11 . . . 4 (𝜑 𝑆 = 𝐴)
43difeq1d 4095 . . 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 10679 . . . 4 (𝜑𝐶 ∈ ℝ*)
95, 6, 8preimalegt 42858 . . 3 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐵𝐶}) = {𝑥𝐴𝐶 < 𝐵})
104, 9eqtr2d 2854 . 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 1906 . . . . . . 7 𝑎 𝐶 ∈ ℝ
1513, 14nfan 1891 . . . . . 6 𝑎(𝜑𝐶 ∈ ℝ)
16 nfv 1906 . . . . . 6 𝑎{𝑥𝐴𝐵𝐶} ∈ 𝑆
1715, 16nfim 1888 . . . . 5 𝑎((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐵𝐶} ∈ 𝑆)
18 eleq1 2897 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ))
1918anbi2d 628 . . . . . 6 (𝑎 = 𝐶 → ((𝜑𝑎 ∈ ℝ) ↔ (𝜑𝐶 ∈ ℝ)))
20 breq2 5061 . . . . . . . 8 (𝑎 = 𝐶 → (𝐵𝑎𝐵𝐶))
2120rabbidv 3478 . . . . . . 7 (𝑎 = 𝐶 → {𝑥𝐴𝐵𝑎} = {𝑥𝐴𝐵𝐶})
2221eleq1d 2894 . . . . . 6 (𝑎 = 𝐶 → ({𝑥𝐴𝐵𝑎} ∈ 𝑆 ↔ {𝑥𝐴𝐵𝐶} ∈ 𝑆))
2319, 22imbi12d 346 . . . . 5 (𝑎 = 𝐶 → (((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆) ↔ ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐵𝐶} ∈ 𝑆)))
24 salpreimalegt.p . . . . 5 ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆)
2517, 23, 24vtoclg1f 3564 . . . 4 (𝐶 ∈ ℝ → ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐵𝐶} ∈ 𝑆))
267, 12, 25sylc 65 . . 3 (𝜑 → {𝑥𝐴𝐵𝐶} ∈ 𝑆)
2711, 26saldifcld 42507 . 2 (𝜑 → ( 𝑆 ∖ {𝑥𝐴𝐵𝐶}) ∈ 𝑆)
2810, 27eqeltrd 2910 1 (𝜑 → {𝑥𝐴𝐶 < 𝐵} ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wnf 1775  wcel 2105  {crab 3139  cdif 3930   cuni 4830   class class class wbr 5057  cr 10524  *cxr 10662   < clt 10663  cle 10664  SAlgcsalg 42470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-xr 10667  df-le 10669  df-salg 42471
This theorem is referenced by:  salpreimalelt  42883  issmfgt  42910
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