Theorem salpreimalegt 47288

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	⊢ Ⅎ𝑥𝜑
salpreimalegt.a	⊢ Ⅎ𝑎𝜑
salpreimalegt.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
salpreimalegt.u	⊢ 𝐴 = ∪ 𝑆
salpreimalegt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
salpreimalegt.p	⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆)
salpreimalegt.c	⊢ (𝜑 → 𝐶 ∈ ℝ)

Assertion

Ref	Expression
salpreimalegt	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 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

Step	Hyp	Ref	Expression
1		salpreimalegt.u	. . . . . 6 ⊢ 𝐴 = ∪ 𝑆
2	1	eqcomi 2773	. . . . 5 ⊢ ∪ 𝑆 = 𝐴
3	2	a1i 11	. . . 4 ⊢ (𝜑 → ∪ 𝑆 = 𝐴)
4	3	difeq1d 4081	. . 3 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) = (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}))
5		salpreimalegt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
6		salpreimalegt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
7		salpreimalegt.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
8	7	rexrd 11234	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
9	5, 6, 8	preimalegt 47279	. . 3 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵})
10	4, 9	eqtr2d 2800	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} = (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}))
11		salpreimalegt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
12	7	ancli 556	. . . 4 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ ℝ))
13		salpreimalegt.a	. . . . . . 7 ⊢ Ⅎ𝑎𝜑
14		nfv 1936	. . . . . . 7 ⊢ Ⅎ𝑎 𝐶 ∈ ℝ
15	13, 14	nfan 1921	. . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝐶 ∈ ℝ)
16		nfv 1936	. . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆
17	15, 16	nfim 1918	. . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆)
18		eleq1 2852	. . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ))
19	18	anbi2d 639	. . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝐶 ∈ ℝ)))
20		breq2 5106	. . . . . . . 8 ⊢ (𝑎 = 𝐶 → (𝐵 ≤ 𝑎 ↔ 𝐵 ≤ 𝐶))
21	20	rabbidv 3423	. . . . . . 7 ⊢ (𝑎 = 𝐶 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} = {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})
22	21	eleq1d 2849	. . . . . 6 ⊢ (𝑎 = 𝐶 → ({𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆))
23	19, 22	imbi12d 346	. . . . 5 ⊢ (𝑎 = 𝐶 → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆) ↔ ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆)))
24		salpreimalegt.p	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆)
25	17, 23, 24	vtoclg1f 3537	. . . 4 ⊢ (𝐶 ∈ ℝ → ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆))
26	7, 12, 25	sylc 65	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆)
27	11, 26	saldifcld 46926	. 2 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) ∈ 𝑆)
28	10, 27	eqeltrd 2864	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562  Ⅎwnf 1805   ∈ wcel 2144  {crab 3416   ∖ cdif 3903  ∪ cuni 4867   class class class wbr 5102  ℝcr 11074  ℝ^*cxr 11217   < clt 11218   ≤ cle 11219  SAlgcsalg 46887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5655  df-cnv 5657  df-xr 11222  df-le 11224  df-salg 46888

This theorem is referenced by:  salpreimalelt  47308  issmfgt  47335