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Theorem hauscmp 20077
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 995 . 2  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  S  C_  X )
2 hauscmp.1 . . . . . 6  |-  X  = 
U. J
3 eqid 2454 . . . . . 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 997 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  J  e.  Haus )
5 simpl2 998 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  S  C_  X
)
6 simpl3 999 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  x  e.  ( X  \  S ) )  ->  ( Jt  S
)  e.  Comp )
7 simpr 459 . . . . . 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 20076 . . . . 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 20002 . . . . . . . . . . 11  |-  ( J  e.  Haus  ->  J  e. 
Top )
1093ad2ant1 1015 . . . . . . . . . 10  |-  ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  ->  J  e.  Top )
11 elssuni 4264 . . . . . . . . . . 11  |-  ( z  e.  J  ->  z  C_ 
U. J )
1211, 2syl6sseqr 3536 . . . . . . . . . 10  |-  ( z  e.  J  ->  z  C_  X )
132sscls 19727 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  z  C_  X )  -> 
z  C_  ( ( cls `  J ) `  z ) )
1410, 12, 13syl2an 475 . . . . . . . . 9  |-  ( ( ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp )  /\  z  e.  J
)  ->  z  C_  ( ( cls `  J
) `  z )
)
15 sstr2 3496 . . . . . . . . 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 563 . . . . . . 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 2929 . . . . . 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 463 . . . . 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 2868 . . 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 19647 . . . 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 19698 . . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808    \ cdif 3458    C_ wss 3461   U.cuni 4235   ` cfv 5570  (class class class)co 6270   ↾t crest 14913   Topctop 19564   Clsdccld 19687   clsccl 19689   Hauscha 19979   Compccmp 20056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-fin 7513  df-fi 7863  df-rest 14915  df-topgen 14936  df-top 19569  df-bases 19571  df-topon 19572  df-cld 19690  df-cls 19692  df-haus 19986  df-cmp 20057
This theorem is referenced by:  txkgen  20322  cmphaushmeo  20470  cnheibor  21624
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