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Theorem iscld 17091
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 17087 . . . 4  |-  ( J  e.  Top  ->  ( Clsd `  J )  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
32eleq2d 2503 . . 3  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  S  e.  { x  e.  ~P X  |  ( X  \  x )  e.  J } ) )
4 difeq2 3459 . . . . 5  |-  ( x  =  S  ->  ( X  \  x )  =  ( X  \  S
) )
54eleq1d 2502 . . . 4  |-  ( x  =  S  ->  (
( X  \  x
)  e.  J  <->  ( X  \  S )  e.  J
) )
65elrab 3092 . . 3  |-  ( S  e.  { x  e. 
~P X  |  ( X  \  x )  e.  J }  <->  ( S  e.  ~P X  /\  ( X  \  S )  e.  J ) )
73, 6syl6bb 253 . 2  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  e.  ~P X  /\  ( X  \  S )  e.  J ) ) )
81topopn 16979 . . . 4  |-  ( J  e.  Top  ->  X  e.  J )
9 elpw2g 4363 . . . 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 686 . 2  |-  ( J  e.  Top  ->  (
( S  e.  ~P X  /\  ( X  \  S )  e.  J
)  <->  ( S  C_  X  /\  ( X  \  S )  e.  J
) ) )
127, 11bitrd 245 1  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  C_  X  /\  ( X 
\  S )  e.  J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709    \ cdif 3317    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   ` cfv 5454   Topctop 16958   Clsdccld 17080
This theorem is referenced by:  iscld2  17092  cldss  17093  cldopn  17095  topcld  17099  discld  17153  indiscld  17155  restcld  17236  ordtcld1  17261  ordtcld2  17262  hauscmp  17470  txcld  17635  ptcld  17645  qtopcld  17745  opnsubg  18137  sszcld  18848  stoweidlem57  27782
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-top 16963  df-cld 17083
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