Theorem salpreimagelt 39595

Description: If all the preimages of left-close, 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	⊢ Ⅎ𝑥𝜑
salpreimagelt.a	⊢ Ⅎ𝑎𝜑
salpreimagelt.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
salpreimagelt.u	⊢ 𝐴 = ∪ 𝑆
salpreimagelt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
salpreimagelt.p	⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆)
salpreimagelt.c	⊢ (𝜑 → 𝐶 ∈ ℝ)

Assertion

Ref	Expression
salpreimagelt	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 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

Step	Hyp	Ref	Expression
1		salpreimagelt.u	. . . . . 6 ⊢ 𝐴 = ∪ 𝑆
2	1	eqcomi 2619	. . . . 5 ⊢ ∪ 𝑆 = 𝐴
3	2	a1i 11	. . . 4 ⊢ (𝜑 → ∪ 𝑆 = 𝐴)
4	3	difeq1d 3689	. . 3 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}))
5		salpreimagelt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
6		salpreimagelt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
7		salpreimagelt.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
8	7	rexrd 9968	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
9	5, 6, 8	preimagelt 39589	. . 3 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶})
10	4, 9	eqtr2d 2645	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} = (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}))
11		salpreimagelt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
12		id 22	. . . . 5 ⊢ (𝜑 → 𝜑)
13	12, 7	jca 553	. . . 4 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ ℝ))
14		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑎𝐶
15		salpreimagelt.a	. . . . . . 7 ⊢ Ⅎ𝑎𝜑
16	14	nfel1 2765	. . . . . . 7 ⊢ Ⅎ𝑎 𝐶 ∈ ℝ
17	15, 16	nfan 1816	. . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝐶 ∈ ℝ)
18		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆
19	17, 18	nfim 1813	. . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆)
20		eleq1 2676	. . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ))
21	20	anbi2d 736	. . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝐶 ∈ ℝ)))
22		breq1 4586	. . . . . . . 8 ⊢ (𝑎 = 𝐶 → (𝑎 ≤ 𝐵 ↔ 𝐶 ≤ 𝐵))
23	22	rabbidv 3164	. . . . . . 7 ⊢ (𝑎 = 𝐶 → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
24	23	eleq1d 2672	. . . . . 6 ⊢ (𝑎 = 𝐶 → ({𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆))
25	21, 24	imbi12d 333	. . . . 5 ⊢ (𝑎 = 𝐶 → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆) ↔ ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆)))
26		salpreimagelt.p	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆)
27	14, 19, 25, 26	vtoclgf 3237	. . . 4 ⊢ (𝐶 ∈ ℝ → ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆))
28	7, 13, 27	sylc 63	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆)
29	11, 28	saldifcld 39241	. 2 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∈ 𝑆)
30	10, 29	eqeltrd 2688	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  {crab 2900   ∖ cdif 3537  ∪ cuni 4372   class class class wbr 4583  ℝcr 9814  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954  SAlgcsalg 39204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-xr 9957  df-le 9959  df-salg 39205

This theorem is referenced by:  salpreimalelt  39615  salpreimagtlt  39616  issmfgelem  39655