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Theorem sscmp 19007
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 18530 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
213ad2ant1 1009 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  ->  J  e.  Top )
3 elpwi 3868 . . . 4  |-  ( x  e.  ~P J  ->  x  C_  J )
4 simpl2 992 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  K  e.  Comp )
5 simprl 755 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  x  C_  J
)
6 simpl3 993 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  J  C_  K
)
75, 6sstrd 3365 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  x  C_  K
)
8 simpl1 991 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  J  e.  (TopOn `  X ) )
9 toponuni 18531 . . . . . . . . 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 756 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  U. J  =  U. x )
1210, 11eqtrd 2474 . . . . . . 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 18991 . . . . . . 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 1218 . . . . . 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 2450 . . . . . . 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 2735 . . . . . 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 210 . . . . 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 615 . . . 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 474 . . 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 2798 . 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 2442 . . 3  |-  U. J  =  U. J
2322iscmp 18990 . 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 664 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  ->  J  e.  Comp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714   E.wrex 2715    i^i cin 3326    C_ wss 3327   ~Pcpw 3859   U.cuni 4090   ` cfv 5417   Fincfn 7309   Topctop 18497  TopOnctopon 18498   Compccmp 18988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-topon 18505  df-cmp 18989
This theorem is referenced by:  kgencmp2  19118  kgen2ss  19127
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