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Theorem clsint2 26350
Description: The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.)
Hypothesis
Ref Expression
clsint2.1  |-  X  = 
U. J
Assertion
Ref Expression
clsint2  |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  ( ( cls `  J ) `  |^| C )  C_  |^|_ c  e.  C  ( ( cls `  J ) `  c ) )
Distinct variable groups:    C, c    J, c    X, c

Proof of Theorem clsint2
StepHypRef Expression
1 sspwuni 4003 . . . 4  |-  ( C 
C_  ~P X  <->  U. C  C_  X )
2 elssuni 3871 . . . . . . . 8  |-  ( c  e.  C  ->  c  C_ 
U. C )
3 sstr2 3199 . . . . . . . 8  |-  ( c 
C_  U. C  ->  ( U. C  C_  X  -> 
c  C_  X )
)
42, 3syl 15 . . . . . . 7  |-  ( c  e.  C  ->  ( U. C  C_  X  -> 
c  C_  X )
)
54adantl 452 . . . . . 6  |-  ( ( J  e.  Top  /\  c  e.  C )  ->  ( U. C  C_  X  ->  c  C_  X
) )
6 intss1 3893 . . . . . . . . 9  |-  ( c  e.  C  ->  |^| C  C_  c )
7 clsint2.1 . . . . . . . . . 10  |-  X  = 
U. J
87clsss 16807 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  c  C_  X  /\  |^| C  C_  c )  -> 
( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
96, 8syl3an3 1217 . . . . . . . 8  |-  ( ( J  e.  Top  /\  c  C_  X  /\  c  e.  C )  ->  (
( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
1093com23 1157 . . . . . . 7  |-  ( ( J  e.  Top  /\  c  e.  C  /\  c  C_  X )  -> 
( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
11103expia 1153 . . . . . 6  |-  ( ( J  e.  Top  /\  c  e.  C )  ->  ( c  C_  X  ->  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) ) )
125, 11syld 40 . . . . 5  |-  ( ( J  e.  Top  /\  c  e.  C )  ->  ( U. C  C_  X  ->  ( ( cls `  J ) `  |^| C )  C_  (
( cls `  J
) `  c )
) )
1312impancom 427 . . . 4  |-  ( ( J  e.  Top  /\  U. C  C_  X )  ->  ( c  e.  C  ->  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) ) )
141, 13sylan2b 461 . . 3  |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  ( c  e.  C  ->  ( ( cls `  J ) `  |^| C )  C_  (
( cls `  J
) `  c )
) )
1514ralrimiv 2638 . 2  |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  A. c  e.  C  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
16 ssiin 3968 . 2  |-  ( ( ( cls `  J
) `  |^| C ) 
C_  |^|_ c  e.  C  ( ( cls `  J
) `  c )  <->  A. c  e.  C  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
1715, 16sylibr 203 1  |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  ( ( cls `  J ) `  |^| C )  C_  |^|_ c  e.  C  ( ( cls `  J ) `  c ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   |^|cint 3878   |^|_ciin 3922   ` cfv 5271   Topctop 16647   clsccl 16771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-cld 16772  df-cls 16774
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