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Theorem hauscmp 17471
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 959 . 2  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  S  C_  X )
2 hauscmp.1 . . . . . 6  |-  X  = 
U. J
3 eqid 2437 . . . . . 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 961 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  J  e.  Haus )
5 simpl2 962 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  S  C_  X
)
6 simpl3 963 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  ( Jt  S
)  e.  Comp )
7 simpr 449 . . . . . 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 17470 . . . . 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 17396 . . . . . . . . . . 11  |-  ( J  e.  Haus  ->  J  e. 
Top )
1093ad2ant1 979 . . . . . . . . . 10  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  J  e.  Top )
11 elssuni 4044 . . . . . . . . . . 11  |-  ( z  e.  J  ->  z  C_ 
U. J )
1211, 2syl6sseqr 3396 . . . . . . . . . 10  |-  ( z  e.  J  ->  z  C_  X )
132sscls 17121 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  z  C_  X )  -> 
z  C_  ( ( cls `  J ) `  z ) )
1410, 12, 13syl2an 465 . . . . . . . . 9  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  z  e.  J
)  ->  z  C_  ( ( cls `  J
) `  z )
)
15 sstr2 3356 . . . . . . . . 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 550 . . . . . . 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 2819 . . . . . 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 453 . . . . 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 2790 . . 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 17041 . . . 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 225 . 2  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  ( X  \  S
)  e.  J )
252iscld 17092 . . 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 890 1  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  S  e.  ( Clsd `  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707   {crab 2710    \ cdif 3318    C_ wss 3321   U.cuni 4016   ` cfv 5455  (class class class)co 6082   ↾t crest 13649   Topctop 16959   Clsdccld 17081   clsccl 17083   Hauscha 17373   Compccmp 17450
This theorem is referenced by:  txkgen  17685  cmphaushmeo  17833  cnheibor  18981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-en 7111  df-dom 7112  df-fin 7114  df-fi 7417  df-rest 13651  df-topgen 13668  df-top 16964  df-bases 16966  df-topon 16967  df-cld 17084  df-cls 17086  df-haus 17380  df-cmp 17451
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