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Theorem mreiincl 14854
Description: A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
mreiincl  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^|_ y  e.  I  S  e.  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.
StepHypRef Expression
1 dfiin2g 4358 . . 3  |-  ( A. y  e.  I  S  e.  C  ->  |^|_ y  e.  I  S  =  |^| { s  |  E. y  e.  I  s  =  S } )
213ad2ant3 1019 . 2  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^|_ y  e.  I  S  =  |^| { s  |  E. y  e.  I  s  =  S } )
3 simp1 996 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  C  e.  (Moore `  X )
)
4 uniiunlem 3588 . . . . 5  |-  ( A. y  e.  I  S  e.  C  ->  ( A. y  e.  I  S  e.  C  <->  { s  |  E. y  e.  I  s  =  S }  C_  C
) )
54ibi 241 . . . 4  |-  ( A. y  e.  I  S  e.  C  ->  { s  |  E. y  e.  I  s  =  S }  C_  C )
653ad2ant3 1019 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  { s  |  E. y  e.  I  s  =  S }  C_  C )
7 n0 3794 . . . . . 6  |-  ( I  =/=  (/)  <->  E. y  y  e.  I )
8 nfra1 2845 . . . . . . . 8  |-  F/ y A. y  e.  I  S  e.  C
9 nfre1 2925 . . . . . . . . . 10  |-  F/ y E. y  e.  I 
s  =  S
109nfab 2633 . . . . . . . . 9  |-  F/_ y { s  |  E. y  e.  I  s  =  S }
11 nfcv 2629 . . . . . . . . 9  |-  F/_ y (/)
1210, 11nfne 2798 . . . . . . . 8  |-  F/ y { s  |  E. y  e.  I  s  =  S }  =/=  (/)
138, 12nfim 1867 . . . . . . 7  |-  F/ y ( A. y  e.  I  S  e.  C  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) )
14 rsp 2830 . . . . . . . . . 10  |-  ( A. y  e.  I  S  e.  C  ->  ( y  e.  I  ->  S  e.  C ) )
1514com12 31 . . . . . . . . 9  |-  ( y  e.  I  ->  ( A. y  e.  I  S  e.  C  ->  S  e.  C ) )
16 elisset 3124 . . . . . . . . . . 11  |-  ( S  e.  C  ->  E. s 
s  =  S )
17 rspe 2922 . . . . . . . . . . . 12  |-  ( ( y  e.  I  /\  E. s  s  =  S )  ->  E. y  e.  I  E. s 
s  =  S )
1817ex 434 . . . . . . . . . . 11  |-  ( y  e.  I  ->  ( E. s  s  =  S  ->  E. y  e.  I  E. s  s  =  S ) )
1916, 18syl5 32 . . . . . . . . . 10  |-  ( y  e.  I  ->  ( S  e.  C  ->  E. y  e.  I  E. s  s  =  S
) )
20 rexcom4 3133 . . . . . . . . . 10  |-  ( E. y  e.  I  E. s  s  =  S  <->  E. s E. y  e.  I  s  =  S )
2119, 20syl6ib 226 . . . . . . . . 9  |-  ( y  e.  I  ->  ( S  e.  C  ->  E. s E. y  e.  I  s  =  S ) )
2215, 21syld 44 . . . . . . . 8  |-  ( y  e.  I  ->  ( A. y  e.  I  S  e.  C  ->  E. s E. y  e.  I  s  =  S ) )
23 abn0 3804 . . . . . . . 8  |-  ( { s  |  E. y  e.  I  s  =  S }  =/=  (/)  <->  E. s E. y  e.  I 
s  =  S )
2422, 23syl6ibr 227 . . . . . . 7  |-  ( y  e.  I  ->  ( A. y  e.  I  S  e.  C  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) ) )
2513, 24exlimi 1859 . . . . . 6  |-  ( E. y  y  e.  I  ->  ( A. y  e.  I  S  e.  C  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) ) )
267, 25sylbi 195 . . . . 5  |-  ( I  =/=  (/)  ->  ( A. y  e.  I  S  e.  C  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) ) )
2726imp 429 . . . 4  |-  ( ( I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) )
28273adant1 1014 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) )
29 mreintcl 14853 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  {
s  |  E. y  e.  I  s  =  S }  C_  C  /\  { s  |  E. y  e.  I  s  =  S }  =/=  (/) )  ->  |^| { s  |  E. y  e.  I  s  =  S }  e.  C
)
303, 6, 28, 29syl3anc 1228 . 2  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^| { s  |  E. y  e.  I  s  =  S }  e.  C )
312, 30eqeltrd 2555 1  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^|_ y  e.  I  S  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2814   E.wrex 2815    C_ wss 3476   (/)c0 3785   |^|cint 4282   |^|_ciin 4326   ` cfv 5588  Moorecmre 14840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-mre 14844
This theorem is referenced by:  mreriincl  14856  mretopd  19399
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