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Theorem mreiincl 16458
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 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝑦𝐼 𝑆𝐶)
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.
StepHypRef Expression
1 dfiin2g 4705 . . 3 (∀𝑦𝐼 𝑆𝐶 𝑦𝐼 𝑆 = {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆})
213ad2ant3 1130 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝑦𝐼 𝑆 = {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆})
3 simp1 1131 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝐶 ∈ (Moore‘𝑋))
4 uniiunlem 3833 . . . . 5 (∀𝑦𝐼 𝑆𝐶 → (∀𝑦𝐼 𝑆𝐶 ↔ {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ⊆ 𝐶))
54ibi 256 . . . 4 (∀𝑦𝐼 𝑆𝐶 → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ⊆ 𝐶)
653ad2ant3 1130 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ⊆ 𝐶)
7 n0 4074 . . . . . 6 (𝐼 ≠ ∅ ↔ ∃𝑦 𝑦𝐼)
8 nfra1 3079 . . . . . . . 8 𝑦𝑦𝐼 𝑆𝐶
9 nfre1 3143 . . . . . . . . . 10 𝑦𝑦𝐼 𝑠 = 𝑆
109nfab 2907 . . . . . . . . 9 𝑦{𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆}
11 nfcv 2902 . . . . . . . . 9 𝑦
1210, 11nfne 3032 . . . . . . . 8 𝑦{𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅
138, 12nfim 1974 . . . . . . 7 𝑦(∀𝑦𝐼 𝑆𝐶 → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅)
14 rsp 3067 . . . . . . . . . 10 (∀𝑦𝐼 𝑆𝐶 → (𝑦𝐼𝑆𝐶))
1514com12 32 . . . . . . . . 9 (𝑦𝐼 → (∀𝑦𝐼 𝑆𝐶𝑆𝐶))
16 elisset 3355 . . . . . . . . . . 11 (𝑆𝐶 → ∃𝑠 𝑠 = 𝑆)
17 rspe 3141 . . . . . . . . . . . 12 ((𝑦𝐼 ∧ ∃𝑠 𝑠 = 𝑆) → ∃𝑦𝐼𝑠 𝑠 = 𝑆)
1817ex 449 . . . . . . . . . . 11 (𝑦𝐼 → (∃𝑠 𝑠 = 𝑆 → ∃𝑦𝐼𝑠 𝑠 = 𝑆))
1916, 18syl5 34 . . . . . . . . . 10 (𝑦𝐼 → (𝑆𝐶 → ∃𝑦𝐼𝑠 𝑠 = 𝑆))
20 rexcom4 3365 . . . . . . . . . 10 (∃𝑦𝐼𝑠 𝑠 = 𝑆 ↔ ∃𝑠𝑦𝐼 𝑠 = 𝑆)
2119, 20syl6ib 241 . . . . . . . . 9 (𝑦𝐼 → (𝑆𝐶 → ∃𝑠𝑦𝐼 𝑠 = 𝑆))
2215, 21syld 47 . . . . . . . 8 (𝑦𝐼 → (∀𝑦𝐼 𝑆𝐶 → ∃𝑠𝑦𝐼 𝑠 = 𝑆))
23 abn0 4097 . . . . . . . 8 ({𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅ ↔ ∃𝑠𝑦𝐼 𝑠 = 𝑆)
2422, 23syl6ibr 242 . . . . . . 7 (𝑦𝐼 → (∀𝑦𝐼 𝑆𝐶 → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅))
2513, 24exlimi 2233 . . . . . 6 (∃𝑦 𝑦𝐼 → (∀𝑦𝐼 𝑆𝐶 → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅))
267, 25sylbi 207 . . . . 5 (𝐼 ≠ ∅ → (∀𝑦𝐼 𝑆𝐶 → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅))
2726imp 444 . . . 4 ((𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅)
28273adant1 1125 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅)
29 mreintcl 16457 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ⊆ 𝐶 ∧ {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅) → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ∈ 𝐶)
303, 6, 28, 29syl3anc 1477 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ∈ 𝐶)
312, 30eqeltrd 2839 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝑦𝐼 𝑆𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wne 2932  wral 3050  wrex 3051  wss 3715  c0 4058   cint 4627   ciin 4673  cfv 6049  Moorecmre 16444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-mre 16448
This theorem is referenced by:  mreriincl  16460  mretopd  21098
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