Theorem mreiincl 17286

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 4966	. . 3 ⊢ (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → ∩ 𝑦 ∈ 𝐼 𝑆 = ∩ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆})
2	1	3ad2ant3 1133	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 = ∩ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆})
3		simp1 1134	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → 𝐶 ∈ (Moore‘𝑋))
4		uniiunlem 4023	. . . . 5 ⊢ (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 ↔ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ⊆ 𝐶))
5	4	ibi 266	. . . 4 ⊢ (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ⊆ 𝐶)
6	5	3ad2ant3 1133	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ⊆ 𝐶)
7		n0 4285	. . . . . 6 ⊢ (𝐼 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐼)
8		nfra1 3144	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶
9		nfre1 3236	. . . . . . . . . 10 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐼 𝑠 = 𝑆
10	9	nfab 2914	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆}
11		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑦∅
12	10, 11	nfne 3046	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅
13	8, 12	nfim 1902	. . . . . . 7 ⊢ Ⅎ𝑦(∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅)
14		rsp 3131	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → (𝑦 ∈ 𝐼 → 𝑆 ∈ 𝐶))
15	14	com12 32	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐼 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → 𝑆 ∈ 𝐶))
16		elisset 2821	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝐶 → ∃𝑠 𝑠 = 𝑆)
17		rspe 3234	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐼 ∧ ∃𝑠 𝑠 = 𝑆) → ∃𝑦 ∈ 𝐼 ∃𝑠 𝑠 = 𝑆)
18	17	ex 412	. . . . . . . . . . 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 3181	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐼 ∃𝑠 𝑠 = 𝑆 ↔ ∃𝑠∃𝑦 ∈ 𝐼 𝑠 = 𝑆)
21	19, 20	syl6ib 250	. . . . . . . . 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 4319	. . . . . . . 8 ⊢ ({𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅ ↔ ∃𝑠∃𝑦 ∈ 𝐼 𝑠 = 𝑆)
24	22, 23	syl6ibr 251	. . . . . . 7 ⊢ (𝑦 ∈ 𝐼 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅))
25	13, 24	exlimi 2213	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝐼 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅))
26	7, 25	sylbi 216	. . . . 5 ⊢ (𝐼 ≠ ∅ → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅))
27	26	imp 406	. . . 4 ⊢ ((𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅)
28	27	3adant1 1128	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅)
29		mreintcl 17285	. . 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 1369	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ∈ 𝐶)
31	2, 30	eqeltrd 2840	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1541  ∃wex 1785   ∈ wcel 2109  {cab 2716   ≠ wne 2944  ∀wral 3065  ∃wrex 3066   ⊆ wss 3891  ∅c0 4261  ∩ cint 4884  ∩ ciin 4930  ‘cfv 6430  Moorecmre 17272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-int 4885  df-iin 4932  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-iota 6388  df-fun 6432  df-fv 6438  df-mre 17276

This theorem is referenced by:  mreriincl  17288  mretopd  22224