Theorem mreiincl 17686

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 4993	. . 3 ⊢ (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → ∩ 𝑦 ∈ 𝐼 𝑆 = ∩ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆})
2	1	3ad2ant3 1153	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 = ∩ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆})
3		simp1 1154	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → 𝐶 ∈ (Moore‘𝑋))
4		uniiunlem 4038	. . . . 5 ⊢ (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 ↔ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ⊆ 𝐶))
5	4	ibi 270	. . . 4 ⊢ (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ⊆ 𝐶)
6	5	3ad2ant3 1153	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ⊆ 𝐶)
7		n0 4303	. . . . . 6 ⊢ (𝐼 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐼)
8		nfra1 3288	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶
9		nfre1 3289	. . . . . . . . . 10 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐼 𝑠 = 𝑆
10	9	nfab 2930	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆}
11		nfcv 2924	. . . . . . . . 9 ⊢ Ⅎ𝑦∅
12	10, 11	nfne 3060	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅
13	8, 12	nfim 1929	. . . . . . 7 ⊢ Ⅎ𝑦(∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅)
14		rsp 3252	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → (𝑦 ∈ 𝐼 → 𝑆 ∈ 𝐶))
15	14	com12 33	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐼 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → 𝑆 ∈ 𝐶))
16		elisset 2844	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝐶 → ∃𝑠 𝑠 = 𝑆)
17		rspe 3254	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐼 ∧ ∃𝑠 𝑠 = 𝑆) → ∃𝑦 ∈ 𝐼 ∃𝑠 𝑠 = 𝑆)
18	17	ex 418	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐼 → (∃𝑠 𝑠 = 𝑆 → ∃𝑦 ∈ 𝐼 ∃𝑠 𝑠 = 𝑆))
19	16, 18	syl5 35	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐼 → (𝑆 ∈ 𝐶 → ∃𝑦 ∈ 𝐼 ∃𝑠 𝑠 = 𝑆))
20		rexcom4 3291	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐼 ∃𝑠 𝑠 = 𝑆 ↔ ∃𝑠∃𝑦 ∈ 𝐼 𝑠 = 𝑆)
21	19, 20	imbitrdi 254	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐼 → (𝑆 ∈ 𝐶 → ∃𝑠∃𝑦 ∈ 𝐼 𝑠 = 𝑆))
22	15, 21	syld 48	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐼 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → ∃𝑠∃𝑦 ∈ 𝐼 𝑠 = 𝑆))
23		abn0 4337	. . . . . . . 8 ⊢ ({𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅ ↔ ∃𝑠∃𝑦 ∈ 𝐼 𝑠 = 𝑆)
24	22, 23	imbitrrdi 255	. . . . . . 7 ⊢ (𝑦 ∈ 𝐼 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅))
25	13, 24	exlimi 2255	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝐼 → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅))
26	7, 25	sylbi 220	. . . . 5 ⊢ (𝐼 ≠ ∅ → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅))
27	26	imp 412	. . . 4 ⊢ ((𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅)
28	27	3adant1 1148	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ≠ ∅)
29		mreintcl 17685	. . 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 1398	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ {𝑠 ∣ ∃𝑦 ∈ 𝐼 𝑠 = 𝑆} ∈ 𝐶)
31	2, 30	eqeltrd 2862	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2740   ≠ wne 2957  ∀wral 3078  ∃wrex 3088   ⊆ wss 3902  ∅c0 4282  ∩ cint 4910  ∩ ciin 4955  ‘cfv 6537  Moorecmre 17672

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-nul 5267  ax-pow 5334  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-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  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-int 4911  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-mre 17676

This theorem is used by:  mreriincl  17688  mretopd  23323