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Theorem clsint2 26247
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 3987 . . . 4  |-  ( C 
C_  ~P X  <->  U. C  C_  X )
2 elssuni 3855 . . . . . . . 8  |-  ( c  e.  C  ->  c  C_ 
U. C )
3 sstr2 3186 . . . . . . . 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 3877 . . . . . . . . 9  |-  ( c  e.  C  ->  |^| C  C_  c )
7 clsint2.1 . . . . . . . . . 10  |-  X  = 
U. J
87clsss 16791 . . . . . . . . 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 2625 . 2  |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  A. c  e.  C  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
16 ssiin 3952 . 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 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   |^|_ciin 3906   ` cfv 5255   Topctop 16631   clsccl 16755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-cld 16756  df-cls 16758
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