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Theorem hauscmp 19773
Description: A compact subspace of a T2 space is closed. (Contributed by Jeff Hankins, 16-Jan-2010.) (Proof shortened by Mario Carneiro, 14-Dec-2013.)
Hypothesis
Ref Expression
hauscmp.1  |-  X  = 
U. J
Assertion
Ref Expression
hauscmp  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  S  e.  ( Clsd `  J ) )

Proof of Theorem hauscmp
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 996 . 2  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  S  C_  X )
2 hauscmp.1 . . . . . 6  |-  X  = 
U. J
3 eqid 2441 . . . . . 6  |-  { y  e.  J  |  E. w  e.  J  (
x  e.  w  /\  ( ( cls `  J
) `  w )  C_  ( X  \  y
) ) }  =  { y  e.  J  |  E. w  e.  J  ( x  e.  w  /\  ( ( cls `  J
) `  w )  C_  ( X  \  y
) ) }
4 simpl1 998 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  J  e.  Haus )
5 simpl2 999 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  S  C_  X
)
6 simpl3 1000 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  ( Jt  S
)  e.  Comp )
7 simpr 461 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  x  e.  ( X  \  S ) )
82, 3, 4, 5, 6, 7hauscmplem 19772 . . . . 5  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  E. z  e.  J  ( x  e.  z  /\  (
( cls `  J
) `  z )  C_  ( X  \  S
) ) )
9 haustop 19698 . . . . . . . . . . 11  |-  ( J  e.  Haus  ->  J  e. 
Top )
1093ad2ant1 1016 . . . . . . . . . 10  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  J  e.  Top )
11 elssuni 4260 . . . . . . . . . . 11  |-  ( z  e.  J  ->  z  C_ 
U. J )
1211, 2syl6sseqr 3533 . . . . . . . . . 10  |-  ( z  e.  J  ->  z  C_  X )
132sscls 19423 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  z  C_  X )  -> 
z  C_  ( ( cls `  J ) `  z ) )
1410, 12, 13syl2an 477 . . . . . . . . 9  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  z  e.  J
)  ->  z  C_  ( ( cls `  J
) `  z )
)
15 sstr2 3493 . . . . . . . . 9  |-  ( z 
C_  ( ( cls `  J ) `  z
)  ->  ( (
( cls `  J
) `  z )  C_  ( X  \  S
)  ->  z  C_  ( X  \  S ) ) )
1614, 15syl 16 . . . . . . . 8  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  z  e.  J
)  ->  ( (
( cls `  J
) `  z )  C_  ( X  \  S
)  ->  z  C_  ( X  \  S ) ) )
1716anim2d 565 . . . . . . 7  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  z  e.  J
)  ->  ( (
x  e.  z  /\  ( ( cls `  J
) `  z )  C_  ( X  \  S
) )  ->  (
x  e.  z  /\  z  C_  ( X  \  S ) ) ) )
1817reximdva 2916 . . . . . 6  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  ( E. z  e.  J  ( x  e.  z  /\  ( ( cls `  J ) `
 z )  C_  ( X  \  S ) )  ->  E. z  e.  J  ( x  e.  z  /\  z  C_  ( X  \  S
) ) ) )
1918adantr 465 . . . . 5  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  ( E. z  e.  J  (
x  e.  z  /\  ( ( cls `  J
) `  z )  C_  ( X  \  S
) )  ->  E. z  e.  J  ( x  e.  z  /\  z  C_  ( X  \  S
) ) ) )
208, 19mpd 15 . . . 4  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  E. z  e.  J  ( x  e.  z  /\  z  C_  ( X  \  S
) ) )
2120ralrimiva 2855 . . 3  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  A. x  e.  ( X  \  S ) E. z  e.  J  ( x  e.  z  /\  z  C_  ( X 
\  S ) ) )
22 eltop2 19343 . . . 4  |-  ( J  e.  Top  ->  (
( X  \  S
)  e.  J  <->  A. x  e.  ( X  \  S
) E. z  e.  J  ( x  e.  z  /\  z  C_  ( X  \  S ) ) ) )
2310, 22syl 16 . . 3  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  ( ( X  \  S )  e.  J  <->  A. x  e.  ( X 
\  S ) E. z  e.  J  ( x  e.  z  /\  z  C_  ( X  \  S ) ) ) )
2421, 23mpbird 232 . 2  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  ( X  \  S
)  e.  J )
252iscld 19394 . . 3  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  C_  X  /\  ( X 
\  S )  e.  J ) ) )
2610, 25syl 16 . 2  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  ( S  e.  (
Clsd `  J )  <->  ( S  C_  X  /\  ( X  \  S )  e.  J ) ) )
271, 24, 26mpbir2and 920 1  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  S  e.  ( Clsd `  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   E.wrex 2792   {crab 2795    \ cdif 3455    C_ wss 3458   U.cuni 4230   ` cfv 5574  (class class class)co 6277   ↾t crest 14690   Topctop 19261   Clsdccld 19383   clsccl 19385   Hauscha 19675   Compccmp 19752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-fin 7518  df-fi 7869  df-rest 14692  df-topgen 14713  df-top 19266  df-bases 19268  df-topon 19269  df-cld 19386  df-cls 19388  df-haus 19682  df-cmp 19753
This theorem is referenced by:  txkgen  20019  cmphaushmeo  20167  cnheibor  21321
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