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Theorem elkgen 17247
Description: Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
elkgen  |-  ( J  e.  (TopOn `  X
)  ->  ( A  e.  (𝑘Gen `  J )  <->  ( A  C_  X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( A  i^i  k
)  e.  ( Jt  k ) ) ) ) )
Distinct variable groups:    A, k    k, J    k, X

Proof of Theorem elkgen
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 kgenval 17246 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  (𝑘Gen `  J
)  =  { x  e.  ~P X  |  A. k  e.  ~P  X
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) } )
21eleq2d 2363 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( A  e.  (𝑘Gen `  J )  <->  A  e.  { x  e.  ~P X  |  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) } ) )
3 ineq1 3376 . . . . . . 7  |-  ( x  =  A  ->  (
x  i^i  k )  =  ( A  i^i  k ) )
43eleq1d 2362 . . . . . 6  |-  ( x  =  A  ->  (
( x  i^i  k
)  e.  ( Jt  k )  <->  ( A  i^i  k )  e.  ( Jt  k ) ) )
54imbi2d 307 . . . . 5  |-  ( x  =  A  ->  (
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) )  <-> 
( ( Jt  k )  e.  Comp  ->  ( A  i^i  k )  e.  ( Jt  k ) ) ) )
65ralbidv 2576 . . . 4  |-  ( x  =  A  ->  ( A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) )  <->  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( A  i^i  k )  e.  ( Jt  k ) ) ) )
76elrab 2936 . . 3  |-  ( A  e.  { x  e. 
~P X  |  A. k  e.  ~P  X
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) }  <->  ( A  e. 
~P X  /\  A. k  e.  ~P  X
( ( Jt  k )  e.  Comp  ->  ( A  i^i  k )  e.  ( Jt  k ) ) ) )
8 toponmax 16682 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
9 elpw2g 4190 . . . . 5  |-  ( X  e.  J  ->  ( A  e.  ~P X  <->  A 
C_  X ) )
108, 9syl 15 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( A  e.  ~P X  <->  A  C_  X
) )
1110anbi1d 685 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( ( A  e.  ~P X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( A  i^i  k )  e.  ( Jt  k ) ) )  <->  ( A  C_  X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( A  i^i  k
)  e.  ( Jt  k ) ) ) ) )
127, 11syl5bb 248 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( A  e.  { x  e.  ~P X  |  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Jt  k ) ) }  <->  ( A  C_  X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( A  i^i  k
)  e.  ( Jt  k ) ) ) ) )
132, 12bitrd 244 1  |-  ( J  e.  (TopOn `  X
)  ->  ( A  e.  (𝑘Gen `  J )  <->  ( A  C_  X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( A  i^i  k
)  e.  ( Jt  k ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   ` cfv 5271  (class class class)co 5874   ↾t crest 13341  TopOnctopon 16648   Compccmp 17129  𝑘Genckgen 17244
This theorem is referenced by:  kgeni  17248  kgentopon  17249  kgenss  17254  kgenidm  17258  iskgen3  17260  kgen2ss  17266  kgencn  17267  kgencn3  17269  txkgen  17362
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-csb 3095  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-ov 5877  df-top 16652  df-topon 16655  df-kgen 17245
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