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Theorem salpreimagelt 43282
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 2831 . . . . 5 𝑆 = 𝐴
32a1i 11 . . . 4 (𝜑 𝑆 = 𝐴)
43difeq1d 4073 . . 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 10680 . . . 4 (𝜑𝐶 ∈ ℝ*)
95, 6, 8preimagelt 43276 . . 3 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶})
104, 9eqtr2d 2858 . 2 (𝜑 → {𝑥𝐴𝐵 < 𝐶} = ( 𝑆 ∖ {𝑥𝐴𝐶𝐵}))
11 salpreimagelt.s . . 3 (𝜑𝑆 ∈ SAlg)
127ancli 552 . . . 4 (𝜑 → (𝜑𝐶 ∈ ℝ))
13 salpreimagelt.a . . . . . . 7 𝑎𝜑
14 nfcv 2979 . . . . . . . 8 𝑎𝐶
1514nfel1 2995 . . . . . . 7 𝑎 𝐶 ∈ ℝ
1613, 15nfan 1900 . . . . . 6 𝑎(𝜑𝐶 ∈ ℝ)
17 nfv 1915 . . . . . 6 𝑎{𝑥𝐴𝐶𝐵} ∈ 𝑆
1816, 17nfim 1897 . . . . 5 𝑎((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐶𝐵} ∈ 𝑆)
19 eleq1 2901 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ))
2019anbi2d 631 . . . . . 6 (𝑎 = 𝐶 → ((𝜑𝑎 ∈ ℝ) ↔ (𝜑𝐶 ∈ ℝ)))
21 breq1 5045 . . . . . . . 8 (𝑎 = 𝐶 → (𝑎𝐵𝐶𝐵))
2221rabbidv 3455 . . . . . . 7 (𝑎 = 𝐶 → {𝑥𝐴𝑎𝐵} = {𝑥𝐴𝐶𝐵})
2322eleq1d 2898 . . . . . 6 (𝑎 = 𝐶 → ({𝑥𝐴𝑎𝐵} ∈ 𝑆 ↔ {𝑥𝐴𝐶𝐵} ∈ 𝑆))
2420, 23imbi12d 348 . . . . 5 (𝑎 = 𝐶 → (((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎𝐵} ∈ 𝑆) ↔ ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐶𝐵} ∈ 𝑆)))
25 salpreimagelt.p . . . . 5 ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎𝐵} ∈ 𝑆)
2618, 24, 25vtoclg1f 3541 . . . 4 (𝐶 ∈ ℝ → ((𝜑𝐶 ∈ ℝ) → {𝑥𝐴𝐶𝐵} ∈ 𝑆))
277, 12, 26sylc 65 . . 3 (𝜑 → {𝑥𝐴𝐶𝐵} ∈ 𝑆)
2811, 27saldifcld 42926 . 2 (𝜑 → ( 𝑆 ∖ {𝑥𝐴𝐶𝐵}) ∈ 𝑆)
2910, 28eqeltrd 2914 1 (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wnf 1785  wcel 2114  {crab 3134  cdif 3905   cuni 4813   class class class wbr 5042  cr 10525  *cxr 10663   < clt 10664  cle 10665  SAlgcsalg 42889
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-cnv 5540  df-xr 10668  df-le 10670  df-salg 42890
This theorem is referenced by:  salpreimalelt  43302  salpreimagtlt  43303  issmfgelem  43341
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