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Theorem sscmp 17148
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 16680 . . 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 3646 . . . 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 3202 . . . . . . 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 16681 . . . . . . . . 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 2328 . . . . . . 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 17132 . . . . . . 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 2304 . . . . . . 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 2577 . . . . . 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 2639 . 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 2296 . . 3  |-  U. J  =  U. J
2322iscmp 17131 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   ` cfv 5271   Fincfn 6879   Topctop 16647  TopOnctopon 16648   Compccmp 17129
This theorem is referenced by:  kgencmp2  17257  kgen2ss  17266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-topon 16655  df-cmp 17130
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