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Theorem iscld 18473
Description: The predicate " S is a closed set." (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
iscld.1  |-  X  = 
U. J
Assertion
Ref Expression
iscld  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  C_  X  /\  ( X 
\  S )  e.  J ) ) )

Proof of Theorem iscld
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . . 5  |-  X  = 
U. J
21cldval 18469 . . . 4  |-  ( J  e.  Top  ->  ( Clsd `  J )  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
32eleq2d 2500 . . 3  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  S  e.  { x  e.  ~P X  |  ( X  \  x )  e.  J } ) )
4 difeq2 3456 . . . . 5  |-  ( x  =  S  ->  ( X  \  x )  =  ( X  \  S
) )
54eleq1d 2499 . . . 4  |-  ( x  =  S  ->  (
( X  \  x
)  e.  J  <->  ( X  \  S )  e.  J
) )
65elrab 3106 . . 3  |-  ( S  e.  { x  e. 
~P X  |  ( X  \  x )  e.  J }  <->  ( S  e.  ~P X  /\  ( X  \  S )  e.  J ) )
73, 6syl6bb 261 . 2  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  e.  ~P X  /\  ( X  \  S )  e.  J ) ) )
81topopn 18361 . . . 4  |-  ( J  e.  Top  ->  X  e.  J )
9 elpw2g 4443 . . . 4  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
108, 9syl 16 . . 3  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
1110anbi1d 697 . 2  |-  ( J  e.  Top  ->  (
( S  e.  ~P X  /\  ( X  \  S )  e.  J
)  <->  ( S  C_  X  /\  ( X  \  S )  e.  J
) ) )
127, 11bitrd 253 1  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  C_  X  /\  ( X 
\  S )  e.  J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755   {crab 2709    \ cdif 3313    C_ wss 3316   ~Pcpw 3848   U.cuni 4079   ` cfv 5406   Topctop 18340   Clsdccld 18462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-iota 5369  df-fun 5408  df-fv 5414  df-top 18345  df-cld 18465
This theorem is referenced by:  iscld2  18474  cldss  18475  cldopn  18477  topcld  18481  discld  18535  indiscld  18537  restcld  18618  ordtcld1  18643  ordtcld2  18644  hauscmp  18852  txcld  19018  ptcld  19028  qtopcld  19128  opnsubg  19520  sszcld  20236  stoweidlem57  29698
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