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Theorem clsint2 26313
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 4168 . . . 4  |-  ( C 
C_  ~P X  <->  U. C  C_  X )
2 elssuni 4035 . . . . . . . 8  |-  ( c  e.  C  ->  c  C_ 
U. C )
3 sstr2 3347 . . . . . . . 8  |-  ( c 
C_  U. C  ->  ( U. C  C_  X  -> 
c  C_  X )
)
42, 3syl 16 . . . . . . 7  |-  ( c  e.  C  ->  ( U. C  C_  X  -> 
c  C_  X )
)
54adantl 453 . . . . . 6  |-  ( ( J  e.  Top  /\  c  e.  C )  ->  ( U. C  C_  X  ->  c  C_  X
) )
6 intss1 4057 . . . . . . . . 9  |-  ( c  e.  C  ->  |^| C  C_  c )
7 clsint2.1 . . . . . . . . . 10  |-  X  = 
U. J
87clsss 17110 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  c  C_  X  /\  |^| C  C_  c )  -> 
( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
96, 8syl3an3 1219 . . . . . . . 8  |-  ( ( J  e.  Top  /\  c  C_  X  /\  c  e.  C )  ->  (
( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
1093com23 1159 . . . . . . 7  |-  ( ( J  e.  Top  /\  c  e.  C  /\  c  C_  X )  -> 
( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
11103expia 1155 . . . . . 6  |-  ( ( J  e.  Top  /\  c  e.  C )  ->  ( c  C_  X  ->  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) ) )
125, 11syld 42 . . . . 5  |-  ( ( J  e.  Top  /\  c  e.  C )  ->  ( U. C  C_  X  ->  ( ( cls `  J ) `  |^| C )  C_  (
( cls `  J
) `  c )
) )
1312impancom 428 . . . 4  |-  ( ( J  e.  Top  /\  U. C  C_  X )  ->  ( c  e.  C  ->  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) ) )
141, 13sylan2b 462 . . 3  |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  ( c  e.  C  ->  ( ( cls `  J ) `  |^| C )  C_  (
( cls `  J
) `  c )
) )
1514ralrimiv 2780 . 2  |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  A. c  e.  C  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
16 ssiin 4133 . 2  |-  ( ( ( cls `  J
) `  |^| C ) 
C_  |^|_ c  e.  C  ( ( cls `  J
) `  c )  <->  A. c  e.  C  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
1715, 16sylibr 204 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 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   |^|cint 4042   |^|_ciin 4086   ` cfv 5446   Topctop 16950   clsccl 17074
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-top 16955  df-cld 17075  df-cls 17077
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