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Theorem clsss 19847
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 3449 . . . . . 6  |-  ( T 
C_  S  ->  ( S  C_  x  ->  T  C_  x ) )
21adantr 463 . . . . 5  |-  ( ( T  C_  S  /\  x  e.  ( Clsd `  J ) )  -> 
( S  C_  x  ->  T  C_  x )
)
32ss2rabdv 3520 . . . 4  |-  ( T 
C_  S  ->  { x  e.  ( Clsd `  J
)  |  S  C_  x }  C_  { x  e.  ( Clsd `  J
)  |  T  C_  x } )
4 intss 4248 . . . 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 17 . . 3  |-  ( T 
C_  S  ->  |^| { x  e.  ( Clsd `  J
)  |  T  C_  x }  C_  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
653ad2ant3 1020 . 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 997 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  J  e.  Top )
8 sstr2 3449 . . . . 5  |-  ( T 
C_  S  ->  ( S  C_  X  ->  T  C_  X ) )
98impcom 428 . . . 4  |-  ( ( S  C_  X  /\  T  C_  S )  ->  T  C_  X )
1093adant1 1015 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  T  C_  X )
11 clscld.1 . . . 4  |-  X  = 
U. J
1211clsval 19830 . . 3  |-  ( ( J  e.  Top  /\  T  C_  X )  -> 
( ( cls `  J
) `  T )  =  |^| { x  e.  ( Clsd `  J
)  |  T  C_  x } )
137, 10, 12syl2anc 659 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( cls `  J
) `  T )  =  |^| { x  e.  ( Clsd `  J
)  |  T  C_  x } )
1411clsval 19830 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
15143adant3 1017 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( cls `  J
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
166, 13, 153sstr4d 3485 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 974    = wceq 1405    e. wcel 1842   {crab 2758    C_ wss 3414   U.cuni 4191   |^|cint 4227   ` cfv 5569   Topctop 19686   Clsdccld 19809   clsccl 19811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-top 19691  df-cld 19812  df-cls 19814
This theorem is referenced by:  ntrss  19848  clsss2  19866  lpsscls  19935  lpss3  19938  cnclsi  20066  cncls  20068  lpcls  20158  cnextcn  20859  clssubg  20899  clsnsg  20900  utopreg  21047  hauseqcn  28330  kur14lem6  29508  clsint2  30557  opnregcld  30558
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