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Theorem clsint2 28665
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 4357 . . . 4  |-  ( C 
C_  ~P X  <->  U. C  C_  X )
2 elssuni 4222 . . . . . . . 8  |-  ( c  e.  C  ->  c  C_ 
U. C )
3 sstr2 3464 . . . . . . . 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 466 . . . . . 6  |-  ( ( J  e.  Top  /\  c  e.  C )  ->  ( U. C  C_  X  ->  c  C_  X
) )
6 intss1 4244 . . . . . . . . 9  |-  ( c  e.  C  ->  |^| C  C_  c )
7 clsint2.1 . . . . . . . . . 10  |-  X  = 
U. J
87clsss 18783 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  c  C_  X  /\  |^| C  C_  c )  -> 
( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
96, 8syl3an3 1254 . . . . . . . 8  |-  ( ( J  e.  Top  /\  c  C_  X  /\  c  e.  C )  ->  (
( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
1093com23 1194 . . . . . . 7  |-  ( ( J  e.  Top  /\  c  e.  C  /\  c  C_  X )  -> 
( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
11103expia 1190 . . . . . 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 440 . . . 4  |-  ( ( J  e.  Top  /\  U. C  C_  X )  ->  ( c  e.  C  ->  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) ) )
141, 13sylan2b 475 . . 3  |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  ( c  e.  C  ->  ( ( cls `  J ) `  |^| C )  C_  (
( cls `  J
) `  c )
) )
1514ralrimiv 2823 . 2  |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  A. c  e.  C  ( ( cls `  J
) `  |^| C ) 
C_  ( ( cls `  J ) `  c
) )
16 ssiin 4321 . 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 369    = wceq 1370    e. wcel 1758   A.wral 2795    C_ wss 3429   ~Pcpw 3961   U.cuni 4192   |^|cint 4229   |^|_ciin 4273   ` cfv 5519   Topctop 18623   clsccl 18747
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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-top 18628  df-cld 18748  df-cls 18750
This theorem is referenced by: (None)
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