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Theorem elkgen 17489
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 17488 . . 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 2454 . 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 3478 . . . . . . 7  |-  ( x  =  A  ->  (
x  i^i  k )  =  ( A  i^i  k ) )
43eleq1d 2453 . . . . . 6  |-  ( x  =  A  ->  (
( x  i^i  k
)  e.  ( Jt  k )  <->  ( A  i^i  k )  e.  ( Jt  k ) ) )
54imbi2d 308 . . . . 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 2669 . . . 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 3035 . . 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 16916 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
9 elpw2g 4304 . . . . 5  |-  ( X  e.  J  ->  ( A  e.  ~P X  <->  A 
C_  X ) )
108, 9syl 16 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( A  e.  ~P X  <->  A  C_  X
) )
1110anbi1d 686 . . 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 249 . 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 245 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   {crab 2653    i^i cin 3262    C_ wss 3263   ~Pcpw 3742   ` cfv 5394  (class class class)co 6020   ↾t crest 13575  TopOnctopon 16882   Compccmp 17371  𝑘Genckgen 17486
This theorem is referenced by:  kgeni  17490  kgentopon  17491  kgenss  17496  kgenidm  17500  iskgen3  17502  kgen2ss  17508  kgencn  17509  kgencn3  17511  txkgen  17605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-top 16886  df-topon 16889  df-kgen 17487
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