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Theorem sscmp 17132
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 16664 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
213ad2ant1 976 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  ->  J  e.  Top )
3 elpwi 3633 . . . 4  |-  ( x  e.  ~P J  ->  x  C_  J )
4 simpl2 959 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  K  e.  Comp )
5 simprl 732 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  x  C_  J
)
6 simpl3 960 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  J  C_  K
)
75, 6sstrd 3189 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  x  C_  K
)
8 simpl1 958 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  J  e.  (TopOn `  X ) )
9 toponuni 16665 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
108, 9syl 15 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  X  =  U. J )
11 simprr 733 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  U. J  =  U. x )
1210, 11eqtrd 2315 . . . . . . 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 17116 . . . . . . 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 1182 . . . . . 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 2291 . . . . . . 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 2564 . . . . . 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 201 . . . . 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 598 . . . 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 460 . . 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 2626 . 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 2283 . . 3  |-  U. J  =  U. J
2322iscmp 17115 . 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 645 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   ` cfv 5255   Fincfn 6863   Topctop 16631  TopOnctopon 16632   Compccmp 17113
This theorem is referenced by:  kgencmp2  17241  kgen2ss  17250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-topon 16639  df-cmp 17114
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