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Theorem mreiincl 14656
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 4314 . . 3  |-  ( A. y  e.  I  S  e.  C  ->  |^|_ y  e.  I  S  =  |^| { s  |  E. y  e.  I  s  =  S } )
213ad2ant3 1011 . 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 988 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  C  e.  (Moore `  X )
)
4 uniiunlem 3551 . . . . 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 1011 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  { s  |  E. y  e.  I  s  =  S }  C_  C )
7 n0 3757 . . . . . 6  |-  ( I  =/=  (/)  <->  E. y  y  e.  I )
8 nfra1 2810 . . . . . . . 8  |-  F/ y A. y  e.  I  S  e.  C
9 nfre1 2891 . . . . . . . . . 10  |-  F/ y E. y  e.  I 
s  =  S
109nfab 2620 . . . . . . . . 9  |-  F/_ y { s  |  E. y  e.  I  s  =  S }
11 nfcv 2616 . . . . . . . . 9  |-  F/_ y (/)
1210, 11nfne 2783 . . . . . . . 8  |-  F/ y { s  |  E. y  e.  I  s  =  S }  =/=  (/)
138, 12nfim 1858 . . . . . . 7  |-  F/ y ( A. y  e.  I  S  e.  C  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) )
14 rsp 2894 . . . . . . . . . 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 3089 . . . . . . . . . . 11  |-  ( S  e.  C  ->  E. s 
s  =  S )
17 rspe 2895 . . . . . . . . . . . 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 3098 . . . . . . . . . 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 3767 . . . . . . . 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 1850 . . . . . 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 1006 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  { s  |  E. y  e.  I  s  =  S }  =/=  (/) )
29 mreintcl 14655 . . 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 1219 . 2  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^| { s  |  E. y  e.  I  s  =  S }  e.  C )
312, 30eqeltrd 2542 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 965    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2439    =/= wne 2648   A.wral 2799   E.wrex 2800    C_ wss 3439   (/)c0 3748   |^|cint 4239   |^|_ciin 4283   ` cfv 5529  Moorecmre 14642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-mre 14646
This theorem is referenced by:  mreriincl  14658  mretopd  18831
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