Theorem salpreimagelt 47522

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	⊢ 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 2771	. . . . 5 ⊢ ∪ 𝑆 = 𝐴
3	2	a1i 11	. . . 4 ⊢ (𝜑 → ∪ 𝑆 = 𝐴)
4	3	difeq1d 4076	. . 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 11286	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
9	5, 6, 8	preimagelt 47514	. . 3 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶})
10	4, 9	eqtr2d 2798	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} = (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}))
11		salpreimagelt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
12	7	ancli 558	. . . 4 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ ℝ))
13		salpreimagelt.a	. . . . . . 7 ⊢ Ⅎ𝑎𝜑
14		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑎𝐶
15	14	nfel1 2940	. . . . . . 7 ⊢ Ⅎ𝑎 𝐶 ∈ ℝ
16	13, 15	nfan 1932	. . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝐶 ∈ ℝ)
17		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆
18	16, 17	nfim 1929	. . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆)
19		eleq1 2850	. . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ))
20	19	anbi2d 642	. . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝐶 ∈ ℝ)))
21		breq1 5110	. . . . . . . 8 ⊢ (𝑎 = 𝐶 → (𝑎 ≤ 𝐵 ↔ 𝐶 ≤ 𝐵))
22	21	rabbidv 3421	. . . . . . 7 ⊢ (𝑎 = 𝐶 → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
23	22	eleq1d 2847	. . . . . 6 ⊢ (𝑎 = 𝐶 → ({𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆))
24	20, 23	imbi12d 347	. . . . 5 ⊢ (𝑎 = 𝐶 → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆) ↔ ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆)))
25		salpreimagelt.p	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆)
26	18, 24, 25	vtoclg1f 3533	. . . 4 ⊢ (𝐶 ∈ ℝ → ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆))
27	7, 12, 26	sylc 66	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆)
28	11, 27	saldifcld 47162	. 2 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∈ 𝑆)
29	10, 28	eqeltrd 2862	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2145  {crab 3414   ∖ cdif 3899  ∪ cuni 4870   class class class wbr 5107  ℝcr 11126  ℝ^*cxr 11269   < clt 11270   ≤ cle 11271  SAlgcsalg 47123

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-xr 11274  df-le 11276  df-salg 47124

This theorem is used by:  salpreimalelt  47544  salpreimagtlt  47545  issmfgelem  47584