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Theorem clsss 18800
Description: Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
clsss  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( cls `  J
) `  T )  C_  ( ( cls `  J
) `  S )
)

Proof of Theorem clsss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sstr2 3474 . . . . . 6  |-  ( T 
C_  S  ->  ( S  C_  x  ->  T  C_  x ) )
21adantr 465 . . . . 5  |-  ( ( T  C_  S  /\  x  e.  ( Clsd `  J ) )  -> 
( S  C_  x  ->  T  C_  x )
)
32ss2rabdv 3544 . . . 4  |-  ( T 
C_  S  ->  { x  e.  ( Clsd `  J
)  |  S  C_  x }  C_  { x  e.  ( Clsd `  J
)  |  T  C_  x } )
4 intss 4260 . . . 4  |-  ( { x  e.  ( Clsd `  J )  |  S  C_  x }  C_  { x  e.  ( Clsd `  J
)  |  T  C_  x }  ->  |^| { x  e.  ( Clsd `  J
)  |  T  C_  x }  C_  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
53, 4syl 16 . . 3  |-  ( T 
C_  S  ->  |^| { x  e.  ( Clsd `  J
)  |  T  C_  x }  C_  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
653ad2ant3 1011 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  |^| { x  e.  ( Clsd `  J
)  |  T  C_  x }  C_  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
7 simp1 988 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  J  e.  Top )
8 sstr2 3474 . . . . 5  |-  ( T 
C_  S  ->  ( S  C_  X  ->  T  C_  X ) )
98impcom 430 . . . 4  |-  ( ( S  C_  X  /\  T  C_  S )  ->  T  C_  X )
1093adant1 1006 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  T  C_  X )
11 clscld.1 . . . 4  |-  X  = 
U. J
1211clsval 18783 . . 3  |-  ( ( J  e.  Top  /\  T  C_  X )  -> 
( ( cls `  J
) `  T )  =  |^| { x  e.  ( Clsd `  J
)  |  T  C_  x } )
137, 10, 12syl2anc 661 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( cls `  J
) `  T )  =  |^| { x  e.  ( Clsd `  J
)  |  T  C_  x } )
1411clsval 18783 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
15143adant3 1008 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( cls `  J
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
166, 13, 153sstr4d 3510 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( cls `  J
) `  T )  C_  ( ( cls `  J
) `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2803    C_ wss 3439   U.cuni 4202   |^|cint 4239   ` cfv 5529   Topctop 18640   Clsdccld 18762   clsccl 18764
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-top 18645  df-cld 18765  df-cls 18767
This theorem is referenced by:  ntrss  18801  clsss2  18818  lpsscls  18887  lpss3  18890  cnclsi  19018  cncls  19020  lpcls  19110  cnextcn  19781  clssubg  19821  clsnsg  19822  utopreg  19969  hauseqcn  26493  kur14lem6  27266  clsint2  28695  opnregcld  28696
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