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Theorem clsss 17110
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 3347 . . . . . 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 3416 . . . 4  |-  ( T 
C_  S  ->  { x  e.  ( Clsd `  J
)  |  S  C_  x }  C_  { x  e.  ( Clsd `  J
)  |  T  C_  x } )
4 intss 4063 . . . 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 3347 . . . . 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 17093 . . 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 17093 . . 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 3383 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 1652    e. wcel 1725   {crab 2701    C_ wss 3312   U.cuni 4007   |^|cint 4042   ` cfv 5446   Topctop 16950   Clsdccld 17072   clsccl 17074
This theorem is referenced by:  ntrss  17111  clsss2  17128  lpsscls  17197  lpss3  17200  cnclsi  17328  cncls  17330  lpcls  17420  cnextcn  18090  clssubg  18130  clsnsg  18131  utopreg  18274  hauseqcn  24285  kur14lem6  24889  clsint2  26323  opnregcld  26324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-top 16955  df-cld 17075  df-cls 17077
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