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Theorem clsss 17034
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 3291 . . . . . 6  |-  ( T 
C_  S  ->  ( S  C_  x  ->  T  C_  x ) )
21adantr 452 . . . . 5  |-  ( ( T  C_  S  /\  x  e.  ( Clsd `  J ) )  -> 
( S  C_  x  ->  T  C_  x )
)
32ss2rabdv 3360 . . . 4  |-  ( T 
C_  S  ->  { x  e.  ( Clsd `  J
)  |  S  C_  x }  C_  { x  e.  ( Clsd `  J
)  |  T  C_  x } )
4 intss 4006 . . . 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 980 . 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 957 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  J  e.  Top )
8 sstr2 3291 . . . . 5  |-  ( T 
C_  S  ->  ( S  C_  X  ->  T  C_  X ) )
98impcom 420 . . . 4  |-  ( ( S  C_  X  /\  T  C_  S )  ->  T  C_  X )
1093adant1 975 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  T  C_  X )
11 clscld.1 . . . 4  |-  X  = 
U. J
1211clsval 17017 . . 3  |-  ( ( J  e.  Top  /\  T  C_  X )  -> 
( ( cls `  J
) `  T )  =  |^| { x  e.  ( Clsd `  J
)  |  T  C_  x } )
137, 10, 12syl2anc 643 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( cls `  J
) `  T )  =  |^| { x  e.  ( Clsd `  J
)  |  T  C_  x } )
1411clsval 17017 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
15143adant3 977 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( cls `  J
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
166, 13, 153sstr4d 3327 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( cls `  J
) `  T )  C_  ( ( cls `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2646    C_ wss 3256   U.cuni 3950   |^|cint 3985   ` cfv 5387   Topctop 16874   Clsdccld 16996   clsccl 16998
This theorem is referenced by:  ntrss  17035  clsss2  17052  lpsscls  17121  lpss3  17124  cnclsi  17251  cncls  17253  lpcls  17343  cnextcn  18012  clssubg  18052  clsnsg  18053  utopreg  18196  hauseqcn  24090  kur14lem6  24669  clsint2  26016  opnregcld  26017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-top 16879  df-cld 16999  df-cls 17001
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