Theorem salpreimalegt 47166

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 2750	. . . . 5 ⊢ ∪ 𝑆 = 𝐴
3	2	a1i 11	. . . 4 ⊢ (𝜑 → ∪ 𝑆 = 𝐴)
4	3	difeq1d 4059	. . 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 11190	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
9	5, 6, 8	preimalegt 47157	. . 3 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵})
10	4, 9	eqtr2d 2777	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} = (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}))
11		salpreimalegt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
12	7	ancli 554	. . . 4 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ ℝ))
13		salpreimalegt.a	. . . . . . 7 ⊢ Ⅎ𝑎𝜑
14		nfv 1922	. . . . . . 7 ⊢ Ⅎ𝑎 𝐶 ∈ ℝ
15	13, 14	nfan 1907	. . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝐶 ∈ ℝ)
16		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆
17	15, 16	nfim 1904	. . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆)
18		eleq1 2829	. . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ))
19	18	anbi2d 637	. . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝐶 ∈ ℝ)))
20		breq2 5079	. . . . . . . 8 ⊢ (𝑎 = 𝐶 → (𝐵 ≤ 𝑎 ↔ 𝐵 ≤ 𝐶))
21	20	rabbidv 3400	. . . . . . 7 ⊢ (𝑎 = 𝐶 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} = {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})
22	21	eleq1d 2826	. . . . . 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 3516	. . . 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 46804	. 2 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) ∈ 𝑆)
28	10, 27	eqeltrd 2841	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548  Ⅎwnf 1791   ∈ wcel 2121  {crab 3393   ∖ cdif 3882  ∪ cuni 4841   class class class wbr 5075  ℝcr 11032  ℝ^*cxr 11173   < clt 11174   ≤ cle 11175  SAlgcsalg 46765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-xp 5627  df-cnv 5629  df-xr 11178  df-le 11180  df-salg 46766

This theorem is referenced by:  salpreimalelt  47186  issmfgt  47213