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Theorem mreiincl 13821
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 4124 . . 3  |-  ( A. y  e.  I  S  e.  C  ->  |^|_ y  e.  I  S  =  |^| { s  |  E. y  e.  I  s  =  S } )
213ad2ant3 980 . 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 957 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  C  e.  (Moore `  X )
)
4 uniiunlem 3431 . . . . 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 233 . . . 4  |-  ( A. y  e.  I  S  e.  C  ->  { s  |  E. y  e.  I  s  =  S }  C_  C )
653ad2ant3 980 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  { s  |  E. y  e.  I  s  =  S }  C_  C )
7 n0 3637 . . . . . 6  |-  ( I  =/=  (/)  <->  E. y  y  e.  I )
8 nfra1 2756 . . . . . . . 8  |-  F/ y A. y  e.  I  S  e.  C
9 nfre1 2762 . . . . . . . . . 10  |-  F/ y E. y  e.  I 
s  =  S
109nfab 2576 . . . . . . . . 9  |-  F/_ y { s  |  E. y  e.  I  s  =  S }
11 nfcv 2572 . . . . . . . . 9  |-  F/_ y (/)
1210, 11nfne 2695 . . . . . . . 8  |-  F/ y { s  |  E. y  e.  I  s  =  S }  =/=  (/)
138, 12nfim 1832 . . . . . . 7  |-  F/ y ( A. y  e.  I  S  e.  C  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) )
14 rsp 2766 . . . . . . . . . 10  |-  ( A. y  e.  I  S  e.  C  ->  ( y  e.  I  ->  S  e.  C ) )
1514com12 29 . . . . . . . . 9  |-  ( y  e.  I  ->  ( A. y  e.  I  S  e.  C  ->  S  e.  C ) )
16 elisset 2966 . . . . . . . . . . 11  |-  ( S  e.  C  ->  E. s 
s  =  S )
17 rspe 2767 . . . . . . . . . . . 12  |-  ( ( y  e.  I  /\  E. s  s  =  S )  ->  E. y  e.  I  E. s 
s  =  S )
1817ex 424 . . . . . . . . . . 11  |-  ( y  e.  I  ->  ( E. s  s  =  S  ->  E. y  e.  I  E. s  s  =  S ) )
1916, 18syl5 30 . . . . . . . . . 10  |-  ( y  e.  I  ->  ( S  e.  C  ->  E. y  e.  I  E. s  s  =  S
) )
20 rexcom4 2975 . . . . . . . . . 10  |-  ( E. y  e.  I  E. s  s  =  S  <->  E. s E. y  e.  I  s  =  S )
2119, 20syl6ib 218 . . . . . . . . 9  |-  ( y  e.  I  ->  ( S  e.  C  ->  E. s E. y  e.  I  s  =  S ) )
2215, 21syld 42 . . . . . . . 8  |-  ( y  e.  I  ->  ( A. y  e.  I  S  e.  C  ->  E. s E. y  e.  I  s  =  S ) )
23 abn0 3646 . . . . . . . 8  |-  ( { s  |  E. y  e.  I  s  =  S }  =/=  (/)  <->  E. s E. y  e.  I 
s  =  S )
2422, 23syl6ibr 219 . . . . . . 7  |-  ( y  e.  I  ->  ( A. y  e.  I  S  e.  C  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) ) )
2513, 24exlimi 1821 . . . . . 6  |-  ( E. y  y  e.  I  ->  ( A. y  e.  I  S  e.  C  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) ) )
267, 25sylbi 188 . . . . 5  |-  ( I  =/=  (/)  ->  ( A. y  e.  I  S  e.  C  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) ) )
2726imp 419 . . . 4  |-  ( ( I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) )
28273adant1 975 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) )
29 mreintcl 13820 . . 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 1184 . 2  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^| { s  |  E. y  e.  I  s  =  S }  e.  C )
312, 30eqeltrd 2510 1  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^|_ y  e.  I  S  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   A.wral 2705   E.wrex 2706    C_ wss 3320   (/)c0 3628   |^|cint 4050   |^|_ciin 4094   ` cfv 5454  Moorecmre 13807
This theorem is referenced by:  mreriincl  13823  mretopd  17156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-mre 13811
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