MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mreiincl Structured version   Visualization version   GIF version

Theorem mreiincl 16858
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 4932 . . 3 (∀𝑦𝐼 𝑆𝐶 𝑦𝐼 𝑆 = {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆})
213ad2ant3 1132 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝑦𝐼 𝑆 = {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆})
3 simp1 1133 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝐶 ∈ (Moore‘𝑋))
4 uniiunlem 4036 . . . . 5 (∀𝑦𝐼 𝑆𝐶 → (∀𝑦𝐼 𝑆𝐶 ↔ {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ⊆ 𝐶))
54ibi 270 . . . 4 (∀𝑦𝐼 𝑆𝐶 → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ⊆ 𝐶)
653ad2ant3 1132 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ⊆ 𝐶)
7 n0 4282 . . . . . 6 (𝐼 ≠ ∅ ↔ ∃𝑦 𝑦𝐼)
8 nfra1 3208 . . . . . . . 8 𝑦𝑦𝐼 𝑆𝐶
9 nfre1 3292 . . . . . . . . . 10 𝑦𝑦𝐼 𝑠 = 𝑆
109nfab 2985 . . . . . . . . 9 𝑦{𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆}
11 nfcv 2979 . . . . . . . . 9 𝑦
1210, 11nfne 3111 . . . . . . . 8 𝑦{𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅
138, 12nfim 1897 . . . . . . 7 𝑦(∀𝑦𝐼 𝑆𝐶 → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅)
14 rsp 3195 . . . . . . . . . 10 (∀𝑦𝐼 𝑆𝐶 → (𝑦𝐼𝑆𝐶))
1514com12 32 . . . . . . . . 9 (𝑦𝐼 → (∀𝑦𝐼 𝑆𝐶𝑆𝐶))
16 elisset 3480 . . . . . . . . . . 11 (𝑆𝐶 → ∃𝑠 𝑠 = 𝑆)
17 rspe 3290 . . . . . . . . . . . 12 ((𝑦𝐼 ∧ ∃𝑠 𝑠 = 𝑆) → ∃𝑦𝐼𝑠 𝑠 = 𝑆)
1817ex 416 . . . . . . . . . . 11 (𝑦𝐼 → (∃𝑠 𝑠 = 𝑆 → ∃𝑦𝐼𝑠 𝑠 = 𝑆))
1916, 18syl5 34 . . . . . . . . . 10 (𝑦𝐼 → (𝑆𝐶 → ∃𝑦𝐼𝑠 𝑠 = 𝑆))
20 rexcom4 3237 . . . . . . . . . 10 (∃𝑦𝐼𝑠 𝑠 = 𝑆 ↔ ∃𝑠𝑦𝐼 𝑠 = 𝑆)
2119, 20syl6ib 254 . . . . . . . . 9 (𝑦𝐼 → (𝑆𝐶 → ∃𝑠𝑦𝐼 𝑠 = 𝑆))
2215, 21syld 47 . . . . . . . 8 (𝑦𝐼 → (∀𝑦𝐼 𝑆𝐶 → ∃𝑠𝑦𝐼 𝑠 = 𝑆))
23 abn0 4308 . . . . . . . 8 ({𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅ ↔ ∃𝑠𝑦𝐼 𝑠 = 𝑆)
2422, 23syl6ibr 255 . . . . . . 7 (𝑦𝐼 → (∀𝑦𝐼 𝑆𝐶 → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅))
2513, 24exlimi 2218 . . . . . 6 (∃𝑦 𝑦𝐼 → (∀𝑦𝐼 𝑆𝐶 → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅))
267, 25sylbi 220 . . . . 5 (𝐼 ≠ ∅ → (∀𝑦𝐼 𝑆𝐶 → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅))
2726imp 410 . . . 4 ((𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅)
28273adant1 1127 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅)
29 mreintcl 16857 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ⊆ 𝐶 ∧ {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ≠ ∅) → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ∈ 𝐶)
303, 6, 28, 29syl3anc 1368 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → {𝑠 ∣ ∃𝑦𝐼 𝑠 = 𝑆} ∈ 𝐶)
312, 30eqeltrd 2914 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝑦𝐼 𝑆𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wex 1781  wcel 2114  {cab 2800  wne 3011  wral 3130  wrex 3131  wss 3908  c0 4265   cint 4851   ciin 4895  cfv 6334  Moorecmre 16844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-mre 16848
This theorem is referenced by:  mreriincl  16860  mretopd  21695
  Copyright terms: Public domain W3C validator