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Theorem sscmp 20031
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 19553 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
213ad2ant1 1017 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  ->  J  e.  Top )
3 elpwi 4024 . . . 4  |-  ( x  e.  ~P J  ->  x  C_  J )
4 simpl2 1000 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  K  e.  Comp )
5 simprl 756 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  x  C_  J
)
6 simpl3 1001 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  J  C_  K
)
75, 6sstrd 3509 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  x  C_  K
)
8 simpl1 999 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  J  e.  (TopOn `  X ) )
9 toponuni 19554 . . . . . . . . 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 757 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K
)  /\  ( x  C_  J  /\  U. J  =  U. x ) )  ->  U. J  =  U. x )
1210, 11eqtrd 2498 . . . . . . 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 20015 . . . . . . 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 1228 . . . . . 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 2459 . . . . . . 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 2968 . . . . . 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 2871 . 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 2457 . . 3  |-  U. J  =  U. J
2322iscmp 20014 . 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   U.cuni 4251   ` cfv 5594   Fincfn 7535   Topctop 19520  TopOnctopon 19521   Compccmp 20012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-topon 19528  df-cmp 20013
This theorem is referenced by:  kgencmp2  20172  kgen2ss  20181
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