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Theorem clsint2 30387
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 4404 . . . 4  |-  ( C 
C_  ~P X  <->  U. C  C_  X )
2 elssuni 4264 . . . . . . . 8  |-  ( c  e.  C  ->  c  C_ 
U. C )
3 sstr2 3496 . . . . . . . 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 464 . . . . . 6  |-  ( ( J  e.  Top  /\  c  e.  C )  ->  ( U. C  C_  X  ->  c  C_  X
) )
6 intss1 4286 . . . . . . . . 9  |-  ( c  e.  C  ->  |^| C  C_  c )
7 clsint2.1 . . . . . . . . . 10  |-  X  = 
U. J
87clsss 19722 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  c  C_  X  /\  |^| C  C_  c )  -> 
( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
96, 8syl3an3 1261 . . . . . . . 8  |-  ( ( J  e.  Top  /\  c  C_  X  /\  c  e.  C )  ->  (
( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
1093com23 1200 . . . . . . 7  |-  ( ( J  e.  Top  /\  c  e.  C  /\  c  C_  X )  -> 
( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
11103expia 1196 . . . . . 6  |-  ( ( J  e.  Top  /\  c  e.  C )  ->  ( c  C_  X  ->  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) ) )
125, 11syld 44 . . . . 5  |-  ( ( J  e.  Top  /\  c  e.  C )  ->  ( U. C  C_  X  ->  ( ( cls `  J ) `  |^| C )  C_  (
( cls `  J
) `  c )
) )
1312impancom 438 . . . 4  |-  ( ( J  e.  Top  /\  U. C  C_  X )  ->  ( c  e.  C  ->  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) ) )
141, 13sylan2b 473 . . 3  |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  ( c  e.  C  ->  ( ( cls `  J ) `  |^| C )  C_  (
( cls `  J
) `  c )
) )
1514ralrimiv 2866 . 2  |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  A. c  e.  C  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
16 ssiin 4365 . 2  |-  ( ( ( cls `  J
) `  |^| C ) 
C_  |^|_ c  e.  C  ( ( cls `  J
) `  c )  <->  A. c  e.  C  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
1715, 16sylibr 212 1  |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  ( ( cls `  J ) `  |^| C )  C_  |^|_ c  e.  C  ( ( cls `  J ) `  c ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   |^|cint 4271   |^|_ciin 4316   ` cfv 5570   Topctop 19561   clsccl 19686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-top 19566  df-cld 19687  df-cls 19689
This theorem is referenced by: (None)
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