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Theorem sscmp 17390
Description: A subset of a compact topology (i.e. a coarser topology) is compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
Hypothesis
Ref Expression
sscmp.1  |-  X  = 
U. K
Assertion
Ref Expression
sscmp  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  ->  J  e.  Comp )

Proof of Theorem sscmp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 16914 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
213ad2ant1 978 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  ->  J  e.  Top )
3 elpwi 3750 . . . 4  |-  ( x  e.  ~P J  ->  x  C_  J )
4 simpl2 961 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  K  e.  Comp )
5 simprl 733 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  x  C_  J
)
6 simpl3 962 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  J  C_  K
)
75, 6sstrd 3301 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  x  C_  K
)
8 simpl1 960 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  J  e.  (TopOn `  X ) )
9 toponuni 16915 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
108, 9syl 16 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  X  =  U. J )
11 simprr 734 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  U. J  =  U. x )
1210, 11eqtrd 2419 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  X  =  U. x )
13 sscmp.1 . . . . . . . 8  |-  X  = 
U. K
1413cmpcov 17374 . . . . . . 7  |-  ( ( K  e.  Comp  /\  x  C_  K  /\  X  = 
U. x )  ->  E. y  e.  ( ~P x  i^i  Fin ) X  =  U. y
)
154, 7, 12, 14syl3anc 1184 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  E. y  e.  ( ~P x  i^i  Fin ) X  =  U. y )
1610eqeq1d 2395 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  ( X  = 
U. y  <->  U. J  = 
U. y ) )
1716rexbidv 2670 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  ( E. y  e.  ( ~P x  i^i 
Fin ) X  = 
U. y  <->  E. y  e.  ( ~P x  i^i 
Fin ) U. J  =  U. y ) )
1815, 17mpbid 202 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  E. y  e.  ( ~P x  i^i  Fin ) U. J  =  U. y )
1918expr 599 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  x  C_  J
)  ->  ( U. J  =  U. x  ->  E. y  e.  ( ~P x  i^i  Fin ) U. J  =  U. y ) )
203, 19sylan2 461 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  x  e.  ~P J )  ->  ( U. J  =  U. x  ->  E. y  e.  ( ~P x  i^i  Fin ) U. J  =  U. y ) )
2120ralrimiva 2732 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  ->  A. x  e.  ~P  J ( U. J  =  U. x  ->  E. y  e.  ( ~P x  i^i  Fin ) U. J  =  U. y ) )
22 eqid 2387 . . 3  |-  U. J  =  U. J
2322iscmp 17373 . 2  |-  ( J  e.  Comp  <->  ( J  e. 
Top  /\  A. x  e.  ~P  J ( U. J  =  U. x  ->  E. y  e.  ( ~P x  i^i  Fin ) U. J  =  U. y ) ) )
242, 21, 23sylanbrc 646 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  ->  J  e.  Comp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650    i^i cin 3262    C_ wss 3263   ~Pcpw 3742   U.cuni 3957   ` cfv 5394   Fincfn 7045   Topctop 16881  TopOnctopon 16882   Compccmp 17371
This theorem is referenced by:  kgencmp2  17499  kgen2ss  17508
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-topon 16889  df-cmp 17372
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