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	⊢ 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 2613	. . . . 5 ⊢ ∪ 𝑆 = 𝐴
3	2	a1i 11	. . . 4 ⊢ (𝜑 → ∪ 𝑆 = 𝐴)
4	3	difeq1d 3683	. . 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 9940	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
9	5, 6, 8	preimalegt 39389	. . 3 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵})
10	4, 9	eqtr2d 2639	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} = (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}))
11		salpreimalegt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
12	7	ancli 571	. . . 4 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ ℝ))
13		nfcv 2745	. . . . 5 ⊢ Ⅎ𝑎𝐶
14		salpreimalegt.a	. . . . . . 7 ⊢ Ⅎ𝑎𝜑
15		nfv 1828	. . . . . . 7 ⊢ Ⅎ𝑎 𝐶 ∈ ℝ
16	14, 15	nfan 1814	. . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝐶 ∈ ℝ)
17		nfv 1828	. . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆
18	16, 17	nfim 1811	. . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆)
19		eleq1 2670	. . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ))
20	19	anbi2d 735	. . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝐶 ∈ ℝ)))
21		breq2 4576	. . . . . . . 8 ⊢ (𝑎 = 𝐶 → (𝐵 ≤ 𝑎 ↔ 𝐵 ≤ 𝐶))
22	21	rabbidv 3158	. . . . . . 7 ⊢ (𝑎 = 𝐶 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} = {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})
23	22	eleq1d 2666	. . . . . 6 ⊢ (𝑎 = 𝐶 → ({𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆))
24	20, 23	imbi12d 332	. . . . 5 ⊢ (𝑎 = 𝐶 → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆) ↔ ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆)))
25		salpreimalegt.p	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆)
26	13, 18, 24, 25	vtoclgf 3231	. . . 4 ⊢ (𝐶 ∈ ℝ → ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆))
27	7, 12, 26	sylc 62	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆)
28	11, 27	saldifcld 39040	. 2 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) ∈ 𝑆)
29	10, 28	eqeltrd 2682	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1474  Ⅎwnf 1698   ∈ wcel 1975  {crab 2894   ∖ cdif 3531  ∪ cuni 4361   class class class wbr 4572  ℝcr 9786  ℝ^*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