Theorem mreiincl 16719

Description: A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
mreiincl	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)

Distinct variable groups:   𝑦,𝐼   𝑦,𝑋   𝑦,𝐶

Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem mreiincl

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiin2g 4821	. . 3 ⊢ (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → ∩ 𝑦 ∈ 𝐼 𝑆 = ∩ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆})
2	1	3ad2ant3 1115	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 = ∩ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆})
3		simp1 1116	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → 𝐶 ∈ (Moore‘𝑋))
4		uniiunlem 3945	. . . . 5 ⊢ (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 ↔ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ⊆ 𝐶))
5	4	ibi 259	. . . 4 ⊢ (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ⊆ 𝐶)
6	5	3ad2ant3 1115	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ⊆ 𝐶)
7		n0 4190	. . . . . 6 ⊢ (𝐼 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐼)
8		nfra1 3163	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶
9		nfre1 3245	. . . . . . . . . 10 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐼 𝑠 = 𝑆
10	9	nfab 2932	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆}
11		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑦∅
12	10, 11	nfne 3064	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅
13	8, 12	nfim 1859	. . . . . . 7 ⊢ Ⅎ𝑦(∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅)
14		rsp 3149	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → (𝑦 ∈ 𝐼 → 𝑆 ∈ 𝐶))
15	14	com12 32	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐼 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → 𝑆 ∈ 𝐶))
16		elisset 3420	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝐶 → ∃𝑠 𝑠 = 𝑆)
17		rspe 3243	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐼 ∧ ∃𝑠 𝑠 = 𝑆) → ∃𝑦 ∈ 𝐼 ∃𝑠 𝑠 = 𝑆)
18	17	ex 405	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐼 → (∃𝑠 𝑠 = 𝑆 → ∃𝑦 ∈ 𝐼 ∃𝑠 𝑠 = 𝑆))
19	16, 18	syl5 34	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐼 → (𝑆 ∈ 𝐶 → ∃𝑦 ∈ 𝐼 ∃𝑠 𝑠 = 𝑆))
20		rexcom4 3190	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐼 ∃𝑠 𝑠 = 𝑆 ↔ ∃𝑠∃𝑦 ∈ 𝐼 𝑠 = 𝑆)
21	19, 20	syl6ib 243	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐼 → (𝑆 ∈ 𝐶 → ∃𝑠∃𝑦 ∈ 𝐼 𝑠 = 𝑆))
22	15, 21	syld 47	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐼 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → ∃𝑠∃𝑦 ∈ 𝐼 𝑠 = 𝑆))
23		abn0 4216	. . . . . . . 8 ⊢ ({𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅ ↔ ∃𝑠∃𝑦 ∈ 𝐼 𝑠 = 𝑆)
24	22, 23	syl6ibr 244	. . . . . . 7 ⊢ (𝑦 ∈ 𝐼 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅))
25	13, 24	exlimi 2147	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝐼 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅))
26	7, 25	sylbi 209	. . . . 5 ⊢ (𝐼 ≠ ∅ → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅))
27	26	imp 398	. . . 4 ⊢ ((𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅)
28	27	3adant1 1110	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅)
29		mreintcl 16718	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ⊆ 𝐶 ∧ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅) → ∩ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ∈ 𝐶)
30	3, 6, 28, 29	syl3anc 1351	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ∈ 𝐶)
31	2, 30	eqeltrd 2860	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ w3a 1068   = wceq 1507  ∃wex 1742   ∈ wcel 2050  {cab 2752   ≠ wne 2961  ∀wral 3082  ∃wrex 3083   ⊆ wss 3823  ∅c0 4172  ∩ cint 4743  ∩ ciin 4787  ‘cfv 6182  Moorecmre 16705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-int 4744  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-iota 6146  df-fun 6184  df-fv 6190  df-mre 16709

This theorem is referenced by:  mreriincl  16721  mretopd  21398