Theorem bj-0int 34011

Description: If 𝐴 is a collection of subsets of 𝑋, like a topology, two equivalent ways to say that arbitrary intersections of elements of 𝐴 relative to 𝑋 belong to some class 𝐵 (in typical applications, 𝐴 itself). (Contributed by BJ, 7-Dec-2021.)

Assertion

Ref	Expression
bj-0int	⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑋

Proof of Theorem bj-0int

Step	Hyp	Ref	Expression
1		ssv 3912	. . . . . . . . 9 ⊢ 𝑋 ⊆ V
2		int0 4796	. . . . . . . . 9 ⊢ ∩ ∅ = V
3	1, 2	sseqtr4i 3925	. . . . . . . 8 ⊢ 𝑋 ⊆ ∩ ∅
4		df-ss 3874	. . . . . . . 8 ⊢ (𝑋 ⊆ ∩ ∅ ↔ (𝑋 ∩ ∩ ∅) = 𝑋)
5	3, 4	mpbi 231	. . . . . . 7 ⊢ (𝑋 ∩ ∩ ∅) = 𝑋
6	5	eqcomi 2804	. . . . . 6 ⊢ 𝑋 = (𝑋 ∩ ∩ ∅)
7	6	eleq1i 2873	. . . . 5 ⊢ (𝑋 ∈ 𝐵 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐵)
8	7	a1i 11	. . . 4 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝑋 ∈ 𝐵 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
9		eldifsn 4626	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ≠ ∅))
10		sstr2 3896	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝒫 𝑋 → 𝑥 ⊆ 𝒫 𝑋))
11		bj-intss 34009	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝒫 𝑋 → (𝑥 ≠ ∅ → ∩ 𝑥 ⊆ 𝑋))
12	10, 11	syl6 35	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝒫 𝑋 → (𝑥 ≠ ∅ → ∩ 𝑥 ⊆ 𝑋)))
13		elpwi 4463	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
14	12, 13	syl11 33	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝑥 ∈ 𝒫 𝐴 → (𝑥 ≠ ∅ → ∩ 𝑥 ⊆ 𝑋)))
15	14	impd 411	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ 𝑋))
16	9, 15	syl5bi 243	. . . . . . 7 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∩ 𝑥 ⊆ 𝑋))
17		df-ss 3874	. . . . . . . . 9 ⊢ (∩ 𝑥 ⊆ 𝑋 ↔ (∩ 𝑥 ∩ 𝑋) = ∩ 𝑥)
18		incom 4099	. . . . . . . . . . 11 ⊢ (∩ 𝑥 ∩ 𝑋) = (𝑋 ∩ ∩ 𝑥)
19	18	eqeq1i 2800	. . . . . . . . . 10 ⊢ ((∩ 𝑥 ∩ 𝑋) = ∩ 𝑥 ↔ (𝑋 ∩ ∩ 𝑥) = ∩ 𝑥)
20		eqcom 2802	. . . . . . . . . 10 ⊢ ((𝑋 ∩ ∩ 𝑥) = ∩ 𝑥 ↔ ∩ 𝑥 = (𝑋 ∩ ∩ 𝑥))
21	19, 20	sylbb 220	. . . . . . . . 9 ⊢ ((∩ 𝑥 ∩ 𝑋) = ∩ 𝑥 → ∩ 𝑥 = (𝑋 ∩ ∩ 𝑥))
22	17, 21	sylbi 218	. . . . . . . 8 ⊢ (∩ 𝑥 ⊆ 𝑋 → ∩ 𝑥 = (𝑋 ∩ ∩ 𝑥))
23		eleq1 2870	. . . . . . . . 9 ⊢ (∩ 𝑥 = (𝑋 ∩ ∩ 𝑥) → (∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵))
24	23	a1i 11	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (∩ 𝑥 = (𝑋 ∩ ∩ 𝑥) → (∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵)))
25	22, 24	syl5 34	. . . . . . 7 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (∩ 𝑥 ⊆ 𝑋 → (∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵)))
26	16, 25	syld 47	. . . . . 6 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → (∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵)))
27	26	ralrimiv 3148	. . . . 5 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵))
28		ralbi 3134	. . . . 5 ⊢ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵) → (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵))
29	27, 28	syl 17	. . . 4 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵))
30	8, 29	anbi12d 630	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ ((𝑋 ∩ ∩ ∅) ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵)))
31	30	biancomd 464	. 2 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ∧ (𝑋 ∩ ∩ ∅) ∈ 𝐵)))
32		0elpw 5147	. . 3 ⊢ ∅ ∈ 𝒫 𝐴
33		inteq 4785	. . . . 5 ⊢ (𝑥 = ∅ → ∩ 𝑥 = ∩ ∅)
34		ineq2 4103	. . . . 5 ⊢ (∩ 𝑥 = ∩ ∅ → (𝑋 ∩ ∩ 𝑥) = (𝑋 ∩ ∩ ∅))
35		eleq1 2870	. . . . 5 ⊢ ((𝑋 ∩ ∩ 𝑥) = (𝑋 ∩ ∩ ∅) → ((𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
36	33, 34, 35	3syl 18	. . . 4 ⊢ (𝑥 = ∅ → ((𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
37	36	bj-raldifsn 34010	. . 3 ⊢ (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ↔ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ∧ (𝑋 ∩ ∩ ∅) ∈ 𝐵)))
38	32, 37	ax-mp 5	. 2 ⊢ (∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ↔ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ∧ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
39	31, 38	syl6bbr 290	1 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081   ≠ wne 2984  ∀wral 3105  Vcvv 3437   ∖ cdif 3856   ∩ cin 3858   ⊆ wss 3859  ∅c0 4211  𝒫 cpw 4453  {csn 4472  ∩ cint 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-nul 5101

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-pw 4455  df-sn 4473  df-uni 4746  df-int 4783

This theorem is referenced by: (None)